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Prof. V. Venkata Rao

CHAPTER 7

Noise Performance of Various Modulation Schemes

7.1 Introduction The process of (electronic) communication becomes quite challenging because of the unwanted electrical signals in a communications system. These undesirable signals, usually termed as noise, are random in nature and interfere with the message signals. The receiver input, in general, consists of (message) signal plus noise, possibly with comparable power levels. The purpose of the receiver is to produce the desired signal with a signal-to-noise ratio that is above a specified value.

In this chapter, we will analyze the noise performance of the modulation schemes discussed in chapters 4 to 6. The results of our analysis will show that, under certain conditions, FM is superior to the linear modulation schemes in combating noise and PCM can provide better signal-to-noise ratio at the receiver output than FM. The trade-offs involved in achieving the superior performance from FM and PCM will be discussed.

We shall begin our study with the noise performance of various CW modulations schemes. In this context, it is the performance of the detector (demodulator) that would be emphasized. We shall first develop a suitable receiver model in which the role of the demodulator is the most important one.

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7.2 Receiver Model and Figure of Merit: Linear Modulation 7.2.1 Receiver model Consider the superheterodyne receiver shown in Fig. 4.75. To study the noise performance we shall make use of simplified model shown in Fig. 7.1. Here, Heq ( f ) is the equivalent IF filter which actually represents the cascade filtering characteristic of the RF, mixer and IF sections of Fig. 4.75. s ( t ) is the desired modulated carrier and w ( t ) represents a sample function of the white Gaussian noise process with the two sided spectral density of

N0 . We treat 2

Heq ( f ) to be an ideal narrowband, bandpass filter, with a passband between fc − W to fc + W for the double sideband modulation schemes. For the case of

SSB, we take the filter passband either between fc − W and fc (LSB) or fc and fc + W (USB). (The transmission bandwidth BT is 2 W for the double sideband

modulation schemes whereas it is W for the case of SSB). Also, in the present context, fc represents the carrier frequency measured at the mixer output; that is fc = fIF .

Fig. 7.1: Receiver model (linear modulation)

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The input to the detector is x ( t ) = s ( t ) + n ( t ) , where n ( t ) is the sample function of a bandlimited (narrowband) white noise process N ( t ) with the power spectral density SN ( f ) =

N0 over the passband of Heq ( f ) . (As Heq ( f ) is 2

treated as a narrowband filter, n ( t ) represents the sample function of a narrowband noise process.)

7.2.2 Figure-of-merit The performance of analog communication systems are measured in terms of Signal-to-Noise Ratio ( SNR ) . The SNR measure is meaningful and unambiguous provided the signal and noise are additive at the point of measurement. We shall define two ( SNR ) quantities, namely, (i) ( SNR )0 and (ii) ( SNR )r .

The output signal-to-noise ratio is defined as,

(SNR )0

=

Average power of the message at receiver output Average noise power at the receiver output

(7.1)

The reference signal-to-noise ratio is defined as,

(SNR )r

⎛ Average power of the modulated ⎞ ⎜ ⎟ message signal at receiver input ⎠ ⎝ = ⎛ Average noise power in the message ⎞ ⎜ ⎟ bandwidth at receiver input ⎝ ⎠

(7.2)

The quantity, ( SNR )r can be viewed as the output signal-to-noise ratio which results from baseband or direct transmission of the message without any modulation as shown in Fig. 7.2. Here, m ( t ) is the baseband message signal with the same power as the modulated wave. For the purpose of comparing different modulation systems, we use the Figure-of-Merit ( FOM ) defined as,

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FOM =

Prof. V. Venkata Rao

(SNR )0 (SNR )r

(7.3)

Fig. 7.2: Ideal Baseband Receiver FOM as defined above provides a normalized ( SNR )0 performance of

the various modulation-demodulation schemes; larger the value of FOM, better is the noise performance of the given communication system.

Before analyzing the SNR performance of various detectors, let us quantify the outputs expected of the (idealized) detectors when the input is a narrowband signal. Let x ( t ) be a real narrowband bandpass signal. From Eq. 1.55, x ( t ) can be expressed as

⎧ xc ( t ) cos ( ωc t ) − xs ( t ) sin ( ωc t ) ⎪ x (t ) = ⎨ ⎪⎩ A ( t ) cos ⎡⎣ωc t + ϕ ( t ) ⎤⎦

( 7.4a ) ( 7.4b )

xc ( t ) and xs ( t ) are the in-phase and quadrature components of x ( t ) . The

envelope A ( t ) and the phase ϕ ( t ) are given by Eq. 1.56. In this chapter, we will analyze the performance of a coherent detector, envelope detector, phase detector and a frequency detector when signals such as x ( t ) are given as input. The outputs of the (idealized) detectors can be expressed mathematically in terms of the quantities involved in Eq. 7.4. These are listed below. (Table 7.1)

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Table 7.1: Outputs from various detectors x ( t ) is input to an ideal Detector output proportional to

i)

Coherent detector

xc ( t )

ii)

Envelope detector

A (t ) ϕ (t )

iii) Phase detector

1 d ϕ (t ) 2π d t

iv) Frequency detector

x ( t ) could be used to represent any of the four types of linear modulated signals

or any one of the two types of angle modulated signals. In fact, x ( t ) could even represent (signal + noise) quantity, as will be seen in the sequel. Table 7.2 gives the quantities xc ( t ) , xs ( t ) , A ( t ) and ϕ ( t ) for the linear and angle modulated signals of Chapter 4 and 5.

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Table 7.2: Components of linear and angle modulated signals Signal 1

DSB-SC Ac m ( t ) cos ( ωc t )

2

xc (t )

x s (t )

A (t )

ϕ (t )

Ac m ( t )

zero

Ac m ( t )

Ac [1 + g m m ( t )]

zero

Ac [1 + g m m ( t )]

0, m ( t ) > 0 π, m (t ) < 0

DSB-LC (AM) Ac [1+ g m m ( t )] cos ( ωc t ) ,

zero

Ac [1 + g m m ( t )] ≥ 0

3

SSB

Ac m ( t ) cos ( ωc t ) 2 A l ± c m ( t ) sin ( ωc t ) 2 4

Ac m ( t ) 2

±

(t ) Ac m

Ac

2

2

m

2

2

l (t ) (t ) + m

tan

−1

(t ) ⎤ ⎡ m ⎢− ⎥ ⎣ m (t ) ⎦

Phase modulation Ac cos [ ωc t + ϕ ( t )] ,

Ac cos ϕ ( t )

Ac sin ϕ ( t )

Ac

Ac cos ϕ ( t )

Ac sin ϕ ( t )

Ac

kp m (t )

ϕ (t ) = kp m (t ) 5

Frequency modulation Ac cos [ ωc t + ϕ ( t )] ,

ϕ ( t ) = 2 π kf

t

∫ m ( τ) d τ

t

2 π kf

∫ m ( τ) d τ

− ∞

− ∞

Example 7.1

Let s ( t ) = Ac cos ( ωm t ) cos ( ωc t ) where fm = 103 Hz and fc = 106 Hz. Let us compute and sketch the output v ( t ) of an ideal frequency detector when s ( t ) is its input.

From Table 7.1, we find that an ideal frequency detector output will be proportional to

1 d ϕ (t ) . For the DSB-SC signal, 2π d t

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⎧⎪0 , m ( t ) > 0 ϕ (t ) = ⎨ ⎪⎩π , m ( t ) < 0

(

)

For the example, m ( t ) = cos ⎡ 2 π × 103 t ⎤ . Hence ϕ ( t ) is shown in Fig. 7.3(b). ⎣ ⎦

Fig. 7.3: (Ideal) frequency detector output of example 7.1 Differentiating the waveform in (b), we obtain v ( t ) , which consists of a sequence of impulses which alternate in polarity, as shown in (c).

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Example 7.2

Let m ( t ) =

1 1 + t2

. Let s ( t ) be an SSB signal with m ( t ) as the message

signal. Assuming that s ( t ) is the input to an ideal ED, let us find the expression for its output v ( t ) .

From Example 1.24, we have l (t ) = m

1 1 + t2 1

2 ⎧ l ( t ) ⎤ 2 ⎫ 2 , we have As the envelope of s ( t ) is ⎨ ⎡⎣ m ( t ) ⎤⎦ + ⎡m ⎣ ⎦ ⎬⎭ ⎩

v (t ) =

1 1 + t2

.

7.3 Coherent Demodulation 7.3.1 DSB-SC The receiver model for coherent detection of DSB-SC signals is shown in Fig. 7.4. The DSB-SC signal is, s ( t ) = Ac m ( t ) cos ( ωc t ) . We assume m ( t ) to be sample function of a WSS process M ( t ) with the power spectral density, SM ( f ) , limited to ± W Hz.

Fig. 7.4: Coherent Detection of DSB-SC.

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The carrier, Ac cos ( ωc t ) , which is independent of the message m ( t ) is actually a sample function of the process Ac cos ( ωc t + Θ ) where Θ is a random variable, uniformly distributed in the interval 0 to 2 π . With the random phase added to the carrier term, Rs ( τ ) , the autocorrelation function of the process S ( t ) (of which s ( t ) is a sample function), is given by, Ac2 Rs ( τ ) = RM ( τ ) cos ( ωc τ ) 2

(7.5a)

where RM ( τ ) is the autocorrelation function of the message process. Fourier transform of Rs ( τ ) yields Ss ( f ) given by, Ss ( f )

Ac2 ⎡SM ( f − fc ) + SM ( f + fc ) ⎤⎦ = 4 ⎣

(7.5b)

Let PM denote the message power, where ∞

PM =

∫

SM ( f ) d f =

−∞

∞

A2 Then, ∫ Ss ( f ) d f = 2 c 4 −∞

W

∫

SM ( f ) d f

−W

fc + W

∫

SM ( f − fc ) d f =

fc − W

Ac2 PM . 2

That is, the average power of the modulated signal s ( t ) is sided) noise power spectral density of message bandwidth 2 W is 2 W ×

Ac2 PM . With the (two 2

N0 , the average noise power in the 2

N0 = W N0 . Hence, 2

Ac2 PM ⎡( SNR ) ⎤ = r ⎦ DSB −SC ⎣ 2 W N0

(7.6)

To arrive at the FOM , we require ( SNR )0 . The input to the detector is x ( t ) = s ( t ) + n ( t ) , where n ( t ) is a narrowband noise quantity. Expressing n ( t )

in terms of its in-phase and quadrature components, we have

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x ( t ) = Ac m ( t ) cos ( ωc t ) + nc ( t ) cos ( ωc t ) − ns sin ( ωc t )

Assuming that the local oscillator output is cos ( ωc t ) , the output v ( t ) of the multiplier in the detector (Fig. 7.4) is given by v (t ) =

1 1 1 Ac m ( t ) + nc ( t ) + ⎡⎣ Ac m ( t ) + nc ( t ) ⎤⎦ cos ( 2 ωc t ) 2 2 2 1 − Ac ns ( t ) sin ( 2 ωc t ) 2

As the LPF rejects the spectral components centered around 2 fc , we have

y (t ) =

1 1 Ac m ( t ) + nc ( t ) 2 2

(7.7)

From Eq. 7.7, we observe that, i)

Signal and noise which are additive at the input to the detector are additive even at the output of the detector

ii)

Coherent detector completely rejects the quadrature component ns ( t ) .

iii)

If the noise spectral density is flat at the detector input over the passband

( fc

− W , fc + W ) , then it is flat over the baseband

(− W , W ) ,

at the

detector output. (Note that nc ( t ) has a flat spectrum in the range − W to

W .) ⎛ 1⎞ As the message component at the output is ⎜ ⎟ Ac m ( t ) , the average ⎝ 2⎠

⎛ A2 ⎞ message power at the output is ⎜ c ⎟ PM . As the spectral density of the in-phase ⎜ 4 ⎟ ⎝ ⎠ noise component is N0 for f ≤ W , the average noise power at the receiver output is

W N0 1 . Therefore, ( 2 W ⋅ N0 ) = 4 2

⎡( SNR ) ⎤ 0 ⎦ DSB −SC ⎣

( A 4) P = 2 c

(W N0 )

M

2

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=

Ac2 PM 2W N0

(7.8)

From Eq. 7.6 and 7.8, we obtain

[FOM ]DSB −SC

=

(SNR )0 (SNR )r

=1

(7.9)

7.3.2 SSB Assuming that LSB has been transmitted, we can write s ( t ) as follows: s (t ) =

Ac A l m ( t ) cos ( ωc t ) + c m ( t ) sin ( ωc t ) 2 2

l ( t ) is the Hilbert transform of m ( t ) . Generalizing, where m

S (t ) =

Ac A l M ( t ) cos ( ωc t ) + c M ( t ) sin ( ωc t ) . 2 2

We can show that the autocorrelation function of S ( t ) , Rs ( τ ) is given by Ac2 ⎡ l M ( τ ) sin ( ω τ ) ⎤ Rs ( τ ) = RM ( τ ) cos ( ωc τ ) + R c ⎦ 4 ⎣ l M ( t ) is the Hilbert transform of R ( t ) . Hence the average signal where R M

power, Rs ( 0 ) = and

(SNR )r

Ac2 PM 4

Ac2 PM = 4W N0

(7.10)

Let n ( t ) = nc ( t ) cos ( ωc t ) − ns ( t ) sin ( ωc t ) (Note that with respect to fc , n ( t ) does not have a locally symmetric spectrum). y (t ) =

1 1 Ac m ( t ) + nc ( t ) 4 2

Hence, the output signal power is

Ac2 PM and the output noise power as 16

⎛ 1⎞ ⎜ 4 ⎟ W N0 . Thus, we obtain, ⎝ ⎠

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(SNR )0,SSB =

Ac2 PM 4 × W N0 16

Ac2 PM = 4W N0

(7.11)

From Eq. 7.10 and 7.11,

( FOM )SSB

=1

(7.12)

From Eq. 7.9 and 7.12, we find that under synchronous detection, SNR performance of DSB-SC and SSB are identical, when both the systems operate with the same signal-to-noise ratio at the input of their detectors.

In arriving at the RHS of Eq. 7.11, we have used the narrowband noise description with respect to fc . We can arrive at the same result, if the noise W ⎛ quantity is written with respect to the centre frequency ⎜ fc − 2 ⎝

⎞ ⎟. ⎠

7.4 Envelope Detection DSB-LC or AM signals are normally envelope detected, though coherent detection can also be used for message recovery. This is mainly because envelope detection is simpler to implement as compared to coherent detection. We shall now compute the ( FOM ) AM . The transmitted signal s ( t ) is given by s ( t ) = Ac ⎡⎣1 + g m m ( t ) ⎤⎦ cos ( ωc t ) 2 Ac2 ⎡1 + g m PM ⎤ ⎣ ⎦ . Hence Then the average signal power in s ( t ) = 2

(SNR )r , DSB −LC

=

(

2 Ac2 1 + gm PM

)

2W N0

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Using the in-phase and quadrature component description of the narrowband noise, the quantity at the envelope detector input, x ( t ) , can be written as x ( t ) = s ( t ) + nc ( t ) cos ( ωc t ) − ns ( t ) sin ( ωc t ) = ⎡⎣ Ac + Ac g m m ( t ) + nc ( t ) ⎤⎦ cos ( ωc t ) − ns ( t ) sin ( ωc t )

(7.14)

The various components of Eq. 7.14 are shown as phasors in Fig. 7.5. The receiver output y ( t ) is the envelope of the input quantity x ( t ) . That is,

{

y ( t ) = ⎡⎣ Ac + Ac g m m ( t ) + nc ( t )⎤⎦ + 2

ns2

}

(t )

1 2

Fig. 7.5: Phasor diagram to analyze the envelope detector

We shall analyze the noise performance of envelope detector for two different cases, namely, (i) large SNR at the detector input and (ii) weak SNR at the detector input.

7.4.1 Large predetection SNR Case (i): If the signal-to-noise ratio at the input to the detector is sufficiently

large, we can approximate y ( t ) as (see Fig. 7.5) y ( t ) ≈ Ac + Ac g m m ( t ) + nc ( t )

(7.15)

On the RHS of Eq. 7.15, there are three quantities: A DC term due to the transmitted carrier, a term proportional to the message and the in-phase noise component. In the final output, the DC is blocked. Hence the average signal

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2 PM . The output noise power being equal power at the output is given by Ac2 g m

to 2W N0 we have,

⎡( SNR ) ⎤ ≈ 0 ⎦ AM ⎣

2 Ac2 g m PM 2W N0

(7.16)

It is to be noted that the signal and noise are additive at the detector output and power spectral density of the output noise is flat over the message bandwidth. From Eq. 7.13 and 7.16 we obtain,

( FOM ) AM

=

2 gm PM

(7.17)

2 1 + gm Pm

As can be seen from Eq. 7.17, the FOM with envelope detection is less than unity. That is, the noise performance of DSB-LC with envelope detection is inferior to that of DSB-SC with coherent detection. Assuming m ( t ) to be a tone signal, Am cos ( ωm t ) and µ = g m Am , simple calculation shows that ( FOM ) AM is

(

µ2

2 + µ2

)

. With the maximum permitted value of µ = 1 , we find that the

( FOM ) AM

is

1 . That is, other factors being equal, DSB-LC has to transmit three 3

times as much power as DSB-SC, to achieve the same quality of noise performance. Of course, this is the price one has to pay for trying to achieve simplicity in demodulation.

7.4.2 Weak predetection SNR In this case, noise term dominates. Let n ( t ) = rn ( t ) cos ⎡⎣ωc t + ψ ( t ) ⎤⎦ . We now construct the phasor diagram using rn ( t ) as the reference phasor (Fig. 7.6). Envelope detector output can be approximated as y ( t ) ≈ rn ( t ) + Ac cos ⎡⎣ ψ ( t ) ⎤⎦ + Ac g m m ( t ) cos ⎡⎣ ψ ( t ) ⎤⎦

(7.18)

From Eq. 7.18, we find that detector output has no term strictly proportional to m ( t ) . The last term on the RHS of Eq. 7.18 contains the message signal m ( t )

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multiplied by the noise quantity, cos ψ ( t ) , which is random; that is, the message

signal is hopelessly mutilated beyond any hope of signal recovery. Also, it is to be noted that signal and noise are no longer additive at the detector output. As such, ( SNR )0 is not meaningful.

Fig. 7.6: Phase diagram to analyze the envelope detector for case (ii)

The mutilation or loss of message at low input SNR is called the threshold effect. That is, there is some value of input SNR , above which the

envelope detector operates satisfactorily whereas if the input SNR falls below this value, performance of the detector deteriorates quite rapidly. Actually, threshold is not a unique point and we may have to use some reasonable criterion in arriving it. Let R denote the random variable obtained by observing the process R ( t ) (of which r ( t ) is a sample function) at some fixed point in time. It is quite reasonable to assume that the detector is operating well into the threshold region if P [R ≥ Ac ] ≥ 0.5 ; where as, if the above probability is 0.01 or less, the detector performance is quite satisfactory. Let us define the quantity, carrier-to-noise ratio, ρ as

ρ =

=

average carrier power average noise power in the transmission bandwidth

Ac2 2 Ac2 = 2W N0 4W N0

We shall now compute the threshold SNR in terms of ρ defined above. As R is Rayleigh variable, we have

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r

fR ( r ) =

2 σN

−

e

r2 2 2 σN

2 = 2W N0 where σN

P [R ≥ Ac ] =

∞

∫ fR ( r ) d r

Ac

= e

−

Ac2 4 W N0

= e− ρ

Solving for ρ from e − ρ = 0.5 , we get ρ = ln 2 = 0.69 or - 1.6 dB. Similarly, from the condition P [R ≥ Ac ] = 0.01, we obtain ρ = ln 100 = 4.6 or 6.6 dB. Based on the above calculations, we state that if ρ ≤ − 1.6 dB, the receiver performance is controlled by the noise and hence its output is not acceptable whereas for ρ ≥ 6.6 dB, the effect of noise is not deleterious. However, reasonable intelligibility and naturalness in voice reception requires a post detection SNR of about 25 dB. That is, for satisfactory reception, we require a value of ρ much greater than what is indicated by the threshold considerations. In other words, additive noise makes the signal quality unacceptable long before multiplicative noise mutilates it. Hence threshold effect is usually not a serious limitation in AM transmission.

We now present two oscilloscope displays of the ED output of an AM signal with tone modulation. They are in flash animation. ED - Display 1: SNR at the input to the detector is about 0 dB. ( m ( t ) is a tone TU

UT

signal at 3 kHz.) Output resembles the sample function of the noise process. Threshold effect is about to be set in. ED - Display 2: SNR at the detector input is about 10 dB. Output of the detector, TU

UT

though resembling fairly closely a tone at 3 kHz, is still not a

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pure tone signal. Some amount of noise is seen riding on the output sine wave and the peaks of the sinewave are not perfectly aligned.

Example 7.3

In a receiver meant for the demodulation of SSB signals, Heq ( f ) has the characteristic shown in Fig. 7.7. Assuming that USB has been transmitted, let us find the FOM of the system.

Fig. 7.7: Heq ( f ) for the Example 7.3 Because of the non-ideal Heq ( f ) , SNc ( f ) will be as shown in Fig. 7.8.

Fig. 7.8: SNc ( f ) of Example 7.3

For SSB with coherent demodulation, we have

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Signal quantity at the output =

Ac m (t ) 4

Noise quantity at the output =

nc ( t ) 2

Output noise power =

=

W

∫

−W

SNc d f

5 N0 W 16

⎛ Ac2 PM ⎞ ⎜⎜ ⎟ 16 ⎟⎠ ⎝ = 5 N W 16 0

(SNR )0

(SNR )r

1 4

=

Ac2 PM 5 N0 W

=

Ac2 PM 4W N0

Hence FOM =

(SNR )0 (SNR )r

=

4 = 0.8 . 5

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Exercise 7.1

In a receiver using coherent demodulation, the equivalent IF filter has the characteristics shown in Fig. 7.9. Compute the output noise power in the range f ≤ 100 Hz assuming N0 = 10− 3 Watts/Hz.

Fig. 7.9: Heq ( f ) for the Exercise 7.1

Ans: 0.225 Watts

Example 7.4

In a laboratory experiment involving envelope detection, AM signal at the input to ED, has the modulation index 0.5 with the carrier amplitude of 2 V. m ( t ) is a tone signal of frequency 5 kHz and fc >> 5 kHz. If the (two-sided) noise PSD at the detector input is 10− 8 Watts/Hz, what is the expected ( SNR )0 of this scheme? By how many dB, this scheme is inferior to DSB-SC?

Spectrum of the AM signal is as shown in Fig. 7.10.

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Fig. 7.10: Spectrum of the AM signal of Example 7.4

(SNR )0, AM As PM =

2 Ac2 gm PM 2W N0

=

2 Am , 2

(SNR )0, AM

Ac2 ( g m Am ) = 4W N0

2

But g m Am = µ = 0.5 . Hence,

(SNR )0, AM

1 4 = 3 4 × 5 × 10 × 2 × 10 − 8 4⋅

=

=

1 40 × 10− 5 104 4

= 36 dB

( FOM ) AM

=

µ2 2 + µ2

( FOM )DSB −SC

=

1 4 2+

1 4

=

1 9

=1

DSB-SC results in an increase in the ( SNR )0 by a factor of 9; that is by 9.54 dB.

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7.5 Receiver Model: Angle Modulation The receiver model to be used in the noise analysis of angle modulated signals is shown in Fig. 7.11. (The block de-emphasis filter is shown with broken lines; the effect of pre-emphasis, de-emphasis will be accounted for subsequently).

Fig. 7.11: Receiver model for the evaluation of noise performance The role of Heq ( f ) is similar to what has been mentioned in the context of Fig. 7.1, with suitable changes in the centre frequency and transmission bandwidth. The centre frequency of the filter is fc = fIF , which for the commercial FM is 10.7 MHz. The bandwidth of the filter is the transmission bandwidth of the angle modulated signals, which is about 200 kHz for the commercial FM. Nevertheless, we treat Heq ( f ) to be a narrowband bandpass filter which passes the signal component s ( t ) without any distortion whereas n ( t ) , the noise component at its output is the sample function of a narrowband noise process with a flat spectrum within the passband. The limiter removes any amplitude variations in the output of the equivalent IF filter. We assume the discriminator to be ideal, responding to either phase variations (phase discriminator) or derivative of the phase variations (frequency discriminator) of the quantity present at its input. The figure of merit ( FOM ) for judging the noise performance is the same as defined in section 7.2.2, namely,

(SNR )0 . (SNR )r

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7.6 Calculation of FOM Let, s ( t ) = Ac cos ⎡⎣ ωc t + ϕ ( t ) ⎤⎦

(7.19)

where ⎧k p m ( t ) , for PM ⎪⎪ t ϕ (t ) = ⎨ ⎪2 π kf ∫ m ( τ ) d τ , for FM −∞ ⎪⎩

⋅⋅⋅

( 7.20a )

⋅⋅⋅

( 7.20b )

The output of Heq ( f ) is, x (t ) = s (t ) + n (t )

(7.21a)

= Ac cos ⎡⎣ ωc t + ϕ ( t ) ⎤⎦ + rn ( t ) cos ⎡⎣ ωc t + ψ ( t ) ⎤⎦

(7.21b)

where, on the RHS of Eq. 7.21(b) we have used the envelope ( rn ( t ) ) and phase

(ψ (t ))

representation of the narrowband noise. As in the case of envelope

detection of AM, we shall consider two cases: i)

Strong predetection SNR , ( Ac >> rn ( t ) most of the time )

ii)

Weak predetection SNR , ( Ac << rn ( t ) most of the time ) .

and

7.6.1 Strong Predetection SNR Consider the phasor diagram shown in Fig. 7.12, where we have used the unmodulated carrier as the reference. r ( t ) represents the envelope of the resultant (signal + noise) phasor and θ ( t ) , the phase angle of the resultant. As far as this analysis is concerned, r ( t ) is of no consequence (any variations in r ( t ) are taken care of by the limiter). We express θ ( t ) as

⎪⎧ rn ( t ) sin ⎣⎡ψ ( t ) − ϕ ( t ) ⎦⎤ ⎪⎫ θ ( t ) = ϕ ( t ) + tan− 1 ⎨ ⎬ ⎪⎩ Ac + rn ( t ) cos ⎡⎣ψ ( t ) − ϕ ( t ) ⎤⎦ ⎪⎭

7.22 Indian Institute of Technology Madras

(7.22a)

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Fig. 7.12: Phasor diagram for the case of strong predetection SNR. If we make the assumption that Ac >> rn ( t ) most of the time, we can write,

θ (t ) ≈ ϕ (t ) +

rn ( t ) sin ⎡⎣ψ ( t ) − ϕ ( t ) ⎤⎦ Ac

Notice that the second term on the RHS of Eq. 7.22(b) has the factor

(7.22b)

rn ( t ) . Thus Ac

when the FM signal is much stronger than the noise, it will suppress the small random phase variations caused by noise; then the FM signal is said to capture the detector. v ( t ) , the output of the discriminator is given by, v ( t ) = kd θ ( t )

=

(phase detector)

kd d θ ( t ) 2π d t

(frequency detector)

where kd is the gain constant of the detector under consideration.

a) Phase Modulation

For PM, ϕ ( t ) = k p m ( t ) . For convenience, let k p kd = 1. Then,

v (t ) ≈ m (t ) +

kd rn ( t ) sin ⎡⎣ψ ( t ) − ϕ ( t ) ⎤⎦ Ac

(7.23)

Again, we treat m ( t ) to be a sample function of a WSS process M ( t ) . Then,

7.23 Indian Institute of Technology Madras

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Prof. V. Venkata Rao

output signal power = PM = M 2 ( t ) = RM ( 0 ) Let

kd rn ( t ) sin ⎡⎣ψ ( t ) − ϕ ( t ) ⎤⎦ Ac

nP ( t ) =

(7.24) (7.25)

To calculate the output noise power, we require the power spectral density of nP ( t ) . This is made somewhat difficult because of ϕ ( t ) in nP ( t ) . The analysis becomes fairly easy if we assume ϕ ( t ) = 0 . Of course, it is possible to derive the PSD of nP ( t ) without making the assumption that ϕ ( t ) = 0 . This has been done in Appendix A7.1. In this appendix, it has been shown that the effect of ϕ ( t ) is to produce spectral components beyond W , which are anyway removed

by the final, LPF. Hence, we proceed with our analysis by setting ϕ ( t ) = 0 on the RHS of Eq. 7.25. Then nP ( t ) reduces to,

kd rn ( t ) sin ⎡⎣ψ ( t ) ⎤⎦ Ac

nP ( t ) = =

kd ns ( t ) Ac

Hence,

But,

2

SNP ( f )

⎛k ⎞ = ⎜ d ⎟ SNs ( f ) ⎝ Ac ⎠

SNs ( f )

B ⎧ ⎪N0 , f ≤ T = ⎨ 2 ⎪⎩ 0 , otherwise

BT is the transmission bandwidth, which for the PM case can be taken as the

value given by Eq. 5.26.

Post detection LPF passes only those spectral components that are within 2

⎛k ⎞ ( − W , W ) . Hence the output noise power = ⎜ d ⎟ 2W N0 , resulting in, ⎝ Ac ⎠

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(SNR )0, PM

=

=

As,

(SNR )r , PM

PM ⎛k ⎞ 2W N0 ⎜ d ⎟ ⎝ Ac ⎠

2

Ac2 k p2 PM 2W N0

(7.26)

(A ) 2 = 2 c

N0 W

we have,

( FOM )PM

=

(SNR )0 (SNR )r

= k p2 PM

(7.27a)

We can express ( FOM )PM in terms of the RMS bandwidth. From Eq. A5.4.7, (Appendix A5.4), we have

( Brms )PM

(

= 2 k p RM ( 0 ) Brms

(

= 2 k p PM Brms

)M

)M

( Brms )PM 2 4 ( Brms ) M 2

Hence k p2 PM =

Using this value in Eq. 7.27(a), we obtain

( FOM )PM

(

)

2

⎡B ⎤ rms PM ⎦ ⎣ = 2 4 ⎡ Brms ⎤ M⎦ ⎣

(

(7.27b)

)

b) Frequency Modulation v (t ) =

kd d θ ( t ) 2π d t

= kf kd m ( t ) +

(7.28a)

kd d ns ( t ) 2 π Ac d t

Again, letting kf kd = 1 , we have

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v (t ) = m (t ) +

kd d ns ( t ) 2 π Ac d t

output signal power = PM Let

kd d ns ( t ) 2 π Ac d t

nF ( t ) =

Then, SNF ( f )

⎛ k ⎞ = ⎜ d ⎟ ⎝ 2 π Ac ⎠

2

j 2πf

2

SNS ( f )

The above step follows from the fact that

d ns ( t ) can be obtained by dt

passing ns ( t ) through a differentiator with the transfer function j 2 π f . Thus,

SNF ( f ) =

kd2 f 2 Ac2

SNS ( f )

Fig. 7.13: Noise spectra at the FM discriminator output

As SNS ( f ) is flat for f ≤

BT , we find that SNF ( f ) is parabolic as shown 2

in Fig. 7.13.

The post detection filter eliminates the spectral components beyond f > W . Hence,

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W

The output noise power =

∫

kd2 f 2 N0 Ac2

−W

df

kd2 N0 ⎛ 2 ⎞ 3 = ⎜ ⎟W Ac2 ⎝ 3 ⎠

(7.29)

This is equal to the hatched area in Fig 7.13.

Again, as in the case of PM, we find that increasing the carrier power has a noise quietening effect. But, of course, there is one major difference between

SNP ( f ) and SNF ( f ) ; namely, the latter is parabolic whereas the former is a fiat spectrum.

The parabolic nature of the output FM noise spectrum implies, that high frequency end of the message spectrum is subject to stronger degradation because of noise. Completing our analysis, we find that

(SNR )0, FM

=

=

3 Ac2 PM

(7.30a)

2 kd2 N0 W 3 3 2 Ac2 PM kf 2 N0 W 3

⎛ Ac2 = ⎜ ⎜ 2N W 0 ⎝

(7.30b)

⎞ 3 kf2 PM ⎟⎟ 2 ⎠ W

(7.30c)

(

)FM . From Appendix A5.4, Eq. A5.4.5,

Let us express ( FOM )FM in terms of Brms

( Brms )FM

RM ( 0 )

= 2 kf

= 2 kf PM That is, kf2 PM

(

)

⎡B ⎤ rms FM ⎦ ⎣ = 4

2

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Hence, ( SNR )0, FM

(

⎡ Ac2 3 ⎢ Brms = 2 N0 W 4 ⎢ W ⎣

)FM ⎤⎥

2

⎥ ⎦

As ( SNR )r is the same as in the case of PM, we have

( FOM )FM

= 3

kf2 PM W2

(

⎡ 3 ⎢ Brms = 4⎢ W ⎣

)FM ⎤⎥

2

(7.31)

⎥ ⎦

For a given peak value of the input signal, we find that the deviation ratio D is proportional to

kf ; hence ( FOM )FM is a quadratic function of D . The price paid W

to achieve a significant value for the FOM is the need for increased transmission bandwidth, BT = 2 ( D + k )W . Of course, we should not forget the fact that the result of Eq. 7.31 is based on the assumption that SNR at the detector input is sufficiently large. How do we justify that increasing D , (that is, the transmission bandwidth), will result in the improvement of the output SNR ? Let us look at Eq. 7.28(b). On the RHS, we have two quantities, namely kf kd m ( t ) and

kd d ns ( t ) . The 2 π Ac d t

latter quantity is dependent only on noise and is independent of the message signal.

kd 2 π Ac

being a constant,

d ns ( t ) is the quantity that causes the dt

perturbation of the instantaneous frequency due to the noise. Let us that assume that it is less than or equal to ( ∆ f )n , most of the time. For a given detector, kd is fixed. Hence, as kf increases, frequency deviation increases, thereby increasing the value of D . Let kf ,2 > kf ,1 . Then, to transmit the same m ( t ) , we require more bandwidth if we use a modulator with the frequency sensitivity kf ,2 instead of kf ,1 . In other words, ( ∆ f )2 = kf ,2 mp > ( ∆ f )1 = kf ,1 mp . Hence,

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Prof. V. Venkata Rao

( ∆ f )n ( ∆ f )2

<

( ∆ f )n ( ∆ f )1

In other words, as the frequency derivation due to the modulating signal keeps increasing, the effect of noise becomes less and less significant , thereby increasing the output SNR .

Example 7.5

A tone of unit amplitude and frequency 600 Hz is sent via FM. The FM receiver has been designed for message signals with a bandwidth upto 1 kHz. The maximum phase deviation produced by the tone is 5 rad. We will show that the ( SNR )0 = 31.3 dB, given that

Ac2 = 105 . 2 N0

From Eq. 7.29, output noise power for a message of bandwidth W is

kd2 N0 ⎛ 2 ⎞ 3 ⎜ ⎟ W . For the problem on hand, W = 1 kHz. Hence output noise Ac2 ⎝ 3 ⎠ power =

kd2

⋅

1 105

3 1000 ) ( ⋅ . We shall assume

3

kf kd = 1 so that kd =

⎡⎣s ( t ) ⎤⎦ = Ac ⎣⎡cos ( ωc t + β sin ωm t ) ⎦⎤ FM This maximum phase deviation produced is β . But

β =

∆f k A = f m. fm fm

As Am = 1, we have 5 = kd =

kf . That is, kf = 3000 . Then, 600 1 . 3000

Output noise power =

1

( 3000 )2

⋅

1 105

7.29 Indian Institute of Technology Madras

⋅

109 3

1 . kf

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Prof. V. Venkata Rao

= Output signal power =

1 2700 1 2

Hence ( SNR )0 = 1350 . = 31.3 dB

Example 7.6

(

)

Compare the FOM of PM and FM when m ( t ) = cos 2 π × 5 × 103 t . The frequency deviation produced in both cases is 50 kHz.

⎛ kp ⎞ ' For the case of PM, we have, ∆ f = ⎜ ⎟ mp ⎝ 2π ⎠ As

m'p = 2 π × 5 × 103 and

kp =

2 π × 50 × 103 2 π × 5 × 103

∆ f = 50 × 103 ,

= 10

Therefore,

( FOM )PM

= k p2 PM =

1 × 100 = 50 2

For the case of FM, ∆ f = kf m p

As mp = 1, we have kf = ∆ f = 50 × 103 Therefore,

( FOM )FM

= 3

kf2 PM W2

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50 × 10 ) ( = 3 (5 × 10 ) 3

3

=

2

1 2

2

3 × 100 = 150 2

The above result shows, that for tone modulation and for a given frequency deviation, FM is superior to PM by a factor of 3. In fact, FM results in

(

superior performance as long as 2 π W mp 7.6 falls under this category.

)

( ) . Evidently, the Example

2

< 3 m'p

2

Example 7.7

(

)

(

)

frequency deviation produced is 50 kHz, find

( FOM )PM . ( FOM )FM

Let m ( t ) = 3 cos 2 π × 103 t + cos 2 π × 5 × 103 t . Assuming that the

m'p = 6 π × 103 + 10 π × 103 = 16 π × 103 mp = 3 + 1 = 4

kp

We have

2π

kp

=

kf

m'p = kf mp = 50 × 103 . That is,

2 π mp 1 = 2 × 103 mp'

Hence,

( FOM )PM ( FOM )FM

1 = 3 =

2

⎛ kp ⎞ 1 1 2 × 25 × 106 ⎜ ⎟ W = × 6 3 4 × 10 ⎝ kf ⎠

25 ≈ 2.1 12

7.31

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This example indicates that PM is superior to FM. It is the PSD of the input signal that decides the superiority or otherwise of the FM over PM. We can gain further insight into this issue by looking at the expressions for the FOM in terms of the RMS bandwidth.

From Eq. 7.27(b) and 7.31, we have

( FOM )PM ( FOM )FM

(

)PM ⎤⎦ W 2 1 2 2 3 ⎡ ⎤ ⎡ ⎤ B B ⎣( rms )M ⎦ ⎣( rms )FM ⎦ ⎡B ⎣ rms

=

2

Assuming the same RMS bandwidth for both PM and FM, we find that PM is superior to FM, if

(

W 2 > 3 ⎡ Brms ⎣

(

If W 2 = 3 ⎡ Brms ⎣

)M ⎤⎦

)M ⎤⎦

2

2

, then both PM and FM result in the same performance.

This case corresponds to the PSD of the message signal, SM ( f ) being uniformly distributed in the range ( − W , W ) . If SM ( f ) decreases with frequency, as it does

(

in most cases of practical interest, then W 2 > 3 ⎡ Brms ⎣

)

2

⎤ and PM is superior M⎦

to FM. This was the situation for the Example 7.7. If, on the other hand, the spectrum

is

(

W 2 < 3 ⎡ Brms ⎣

more

)M ⎤⎦

heavily

weighted

at

the

higher

frequencies,

then

2

, and FM gives rise to better performance. This was the

situation for the Example 7.6, where the entire spectrum was concentrated at the tail end (at 5 kHz) with nothing in between.

In most of the real world information bearing signals, such as voice, music etc. have spectral behavior that tapers off with increase in frequency. Then, why not have PM broadcast than FM transmission? As will be seen in the context of pre-emphasis and de-emphasis in FM, the so called FM transmission is really a

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combination of PM and FM, resulting in a performance which is better than either PM or FM alone.

We have developed two different criteria for comparing the SNR performance of PM and FM, namely, PM is superior to FM, if either 3 m'p

C1)

W2 >

C2)

W 2 > 3 ⎡ Brms ⎣

4 π2 mp2

(

, or

)M ⎤⎦

2

is satisfied.

Then which criterion is to be used in practice? C1 is based on the transmission bandwidth where as C2 is based on the RMS bandwidth. Though C1 is generally preferred, in most cases of practical interest, it may be difficult to arrive at the parameters required for C1. Then the only way to make comparison is through C2.

7.6.2 Weak predetection SNR: Threshold effect Consider the phasor diagram shown in Fig. 7.14, where the noise phasor is of a much larger magnitude, compared to the carrier phasor. Then, θ ( t ) can be approximated as

θ (t ) ≈ ψ (t ) +

Ac sin ⎡⎣ϕ ( t ) − ψ ( t ) ⎤⎦ rn ( t )

7.33 Indian Institute of Technology Madras

(7.32)

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Fig. 7.14: Phasor diagram for the case of weak predetection SNR As can be seen from Eq. 7.32, there is no term in θ ( t ) that represents only the signal quantity; the term that contains the signal quantity in θ ( t ) is actually multiplied by

Ac , which is random. This situation is somewhat rn ( t )

analogous to the envelope detection of AM with low predetection SNR . Thus, we can expect a threshold effect in the case of FM demodulation as well. As

Ac , rn ( t )

is small most of the time, phase of r ( t ) is essentially decided by ψ ( t ) . As ψ ( t ) is uniformly distributed, it is quite likely that in short time intervals such as ( t1 , t2 ) ,

( t3 , t 4 )

etc., θ ( t ) changes by 2 π (i.e., r ( t ) rotates around the origin) as shown

in Fig. 7.15(a).

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Fig. 7.15: Occurrence of short pulses at the frequency discriminator output for low predetection SNR.

When such phase variations go through a circuit responding to

dθ , a series of dt

short pulses appear at the output (Fig. 7.15(b)). The duration and frequency (average number of pulses per unit time) of such pulses will depend on the predetection SNR . If SNR is quite low, the frequency of the pulses at the discriminator output increases. As these short pulses have enough energy at the low frequencies, they give rise to crackling or sputtering sound at the receiver (speaker) output. The ( SNR )0 formula derived earlier, for the large input SNR case is no longer valid. As the input SNR keeps decreasing, it is even

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meaningless to talk of ( SNR )0 . In such a situation, the receiver is captured by noise and is said to be working in the threshold region. (To gain some insight into the occurrence of the threshold phenomenon, let us perform the following experiment. An unmodulated sinewave + bandlimited white noise is applied as input to an FM discriminator. The frequency of the sinusoid can be set to the centre frequency of the discriminator and the PSD of the noise is symmetrical with respect to the frequency of the sinusoid. To start with, the input SNR is made very high. If the discriminator output is observed on an oscilloscope, it may resemble the sample function of a bandlimited white noise. As the noise power is increased, impulses start appearing in the output. The input SNR value at which these spikes or impulses start appearing is indicative of the setting in of the threshold behavior).

We now present a few oscilloscopic displays of the experiment suggested above. Display-1 and Display-2 are in flash animation. FM: Display - 1: (Carrier + noise) at the input to PLL with a Carrier-to-Noise Ratio TU

UT

(CNR) of about 15 dB. FM: Display - 2: Output of the PLL for the above input. Note that the response of TU

UT

the PLL to a signal at the carrier frequency is zero. Hence, display-2 is the response of the PLL for the noise input which again looks like a noise waveform. FM: Display - 3: Expanded version of a small part of display - 2. This could be treated as a part of a sample function of the output noise process. FM: Display - 4: (Carrier + noise) at the input to the PLL. CNR is 0 dB. The effect of noise is more prominent in this display when compared to the 15 dB case. FM: Display - 5: Output of the PLL with the input corresponding to 0 dB CNR. Appearance of spikes is clearly evident.

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FM: Display - 3

FM: Display - 4

FM: Display - 5

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As in the case of AM noise analysis, if we set the limit that for the FM detector to operate above the threshold as, P [Rn > Ac ] ≤ 0.01 , then we find that the minimum carrier-to-noise ratio ρ =

Ac2 required is about 5. But, 2 BT N0

experimental results indicate that to obtain the predicted SNR improvement of Ac2 > 20 BT N0 , then the the WBFM, ρ is of the order of 20, or 13 dB. That is, if 2

FM detector will be free from the threshold effect.

Fig. 7.16 gives the plots of ( SNR )0 vs. ( SNR )r

Ac2 = for the case of 2 N0 W

FM with tone modulation. If we take BT = 2 ( β + 1)W , threshold value of ( SNR )r will approximately be ⎡⎣13 + 10 log ( β + 1) ⎤⎦ dB. More details on the threshold effect in FM can be found in [1, 2].

Fig. 7.16: ( SNR )0 performance of WBFM

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For the FM demodulator operating above the threshold, we have (Eq. 7.31),

(SNR )0 (SNR )r

=

3 kf2 Pm W2

For a tone signal, Am cos ( ωm t ) , Pm = β =

2 Am , W = fm and kf Am = ∆ f . As 2

∆f , we have fm

(SNR )0 (SNR )r

=

3 2 β . 2

That is,

⎛ 3 β2 ⎞ ⎡ ⎤ ⎡ ⎤ = 10log10 ⎣( SNR )r ⎦ + 10log10 ⎜ 10log10 ⎣( SNR )0 ⎦ ⎜ 2 ⎟⎟ FM FM ⎝ ⎠

(7.33)

That is, WBFM operating above threshold provides an improvement of

⎛ 3 β2 ⎞ 10log10 ⎜ dB, with respect to ⎜ 2 ⎟⎟ ⎝ ⎠

(SNR )r .

For β = 2 , this amounts to an

improvement of about 7.7 dB and β = 5 , the improvement is about 15.7 dB. This is evident from the plots in Fig. 7.16.

We make a few observations with respect to the plots in Fig. 7.16. (i)

Above threshold [i.e. ( SNR )r above the knee for each curve), WBFM gives rise to impressive ( SNR )0 performance when compared to DSB-SC or SSB with coherent detection. For the latter, ( SNR )0 is best equal to ( SNR )r . (Using pre-emphasis and de-emphasis, the performance of FM can be improved further).

(ii)

Simply increasing the bandwidth without a corresponding increase in the transmitted power does not improve

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because of the threshold

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effect. For example with ( SNR )r about 18dB, β = 2 and β = 5 give rise to the same kind of performance. If ( SNR )r is reduced a little, say to about 16dB, the

(SNR )0

performance with β = 5 is much inferior to that of

β = 2.

7.7 Pre-Emphasis and De-Emphasis in FM For many signals of common interest, such as speech, music etc., most of the energy concentration is in the low frequencies and the frequency components near about W have very little energy in them. When these low energy, highfrequency components frequency modulate a carrier, they will not give rise to full frequency deviation and hence the message will not be utilizing fully the allocated bandwidth. Unfortunately, as was established in the previous section, the noise PSD at the discriminator output increases as f 2 . The net result is an unacceptably low SNR at the high frequency end of the message spectrum. Nevertheless, proper reproduction of the high frequency (but low energy) spectral components of the input spectrum becomes essential from the point of view of final tonal quality or aesthetic appeal. To offset this undesirable occurrence, a clever but easy-to-implement signal processing scheme has been proposed which is popularly known as pre-emphasis and de-emphasis technique.

Pre-emphasis consists in artificially boosting the spectral components in the latter part of the message spectrum. This is accomplished by passing the message signal m ( t ) , through a filter called the pre-emphasis filter, denoted HPE ( f ) . The pre-emphasized signal is used to frequency modulate the carrier at

the transmitting end. In the receiver, the inverse operation, de-emphasis, is performed. This is accomplished by passing the discriminator output through a filter, called the de-emphasis filter, denoted HDE ( f ) . (See Fig. 7.11.) The de-

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emphasis operation will restore all the spectral components of m ( t ) to their original level; this implies the attenuation of the high frequency end of the demodulated spectrum. In this process, the high frequency noise components are also attenuated, thereby improving the overall SNR at the receiver output. Let SNF ( f ) denote the PSD of the noise at the discriminator output. Then the noise power spectral density at the output of the de-emphasis filter is

HDE ( f ) SNF ( f ) . Hence, 2

W

Output noise power with de-emphasis =

∫

−W

HDE ( f ) SNF ( f ) d f 2

(7.34)

As the message power is unaffected because of PE-DE operations.

⎛ 1 ⎞ ⎜⎜ Note that HDE ( f ) = ⎟ , it follows that the improvement in the output HPE ( f ) ⎟⎠ ⎝ SNR is due to the reduced noise power after de-emphasis. We quantify the

improvement in output SNR , produced by PE-DE operation by the improvement

factor I , where I =

average output noise power without PE - DE average output noise power with PE - DE

The numerator of Eq. 7.35 is

2 kd2 N0 W 3 3 Ac2

(7.35)

. As the frequency range of interest is

only f ≤ W , let us take

SNF ( f )

⎧ 2 kd2 N0 f 2 , f ≤ W ⎪ = ⎨ Ac2 ⎪ 0 , otherwise ⎩

We can now compute the denominator of Eq. 7.35 and thereby the improvement factor, which is given by

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2W 3

I =

W

3

∫

(7.36)

f HDE ( f ) d f 2

2

−W

We shall now describe the commonly used PE-DE networks in the commercial FM broadcast and calculate the corresponding improvement in output SNR .

Fig 7.17(a) gives the PE network and 7.17(b), the corresponding DE network used in commercial FM broadcast. In terms of the Laplace transform,

HPE ( s ) =

1 rC K1 R+r s+ r RC

s+

(7.37)

where K1 is a constant to be chosen appropriately. Usually R << r . Hence,

HPE ( s ) ≈

1 rC K1 1 s+ RC

s+

(7.38)

Fig. 7.17: Circuit schematic of a PE-DE network

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The time constant TC1 = r C normally is 75 µ sec. lf ω1 = 2 π f1 =

1 , TC1

then f1 = 2.1 kHz1. The value of TC 2 = R C is not very critical, provided TP

f2 =

PT

1 is not less than the highest audio frequency for which pre-emphasis 2 πTC 2

is desired (15 kHz).

Bode plots for the PE and DE networks are given in Fig. 7.18. Eq. 7.38 can be written as

HPE ( s ) ≈

RC rC

K1

with s = j 2 π f , HPE ( f ) ≈

K =

where

R r

1+ srC 1 + s RC

(7.39a)

⎛f ⎞ 1+ j⎜ ⎟ ⎝ f1 ⎠ K ⎛f ⎞ 1+ j⎜ ⎟ ⎝ f2 ⎠

(7.39b)

K1 .

For f ≤ f2 , we can take

HPE ( f ) =

Hence HDE ( f ) =

1 TP

PT

⎡ ⎛ f ⎞⎤ K ⎢1 + j ⎜ ⎟ ⎥ ⎝ f1 ⎠ ⎦ ⎣ 1 K

⎡ ⎢1 + ⎣

⎛ f ⎞⎤ j ⎜ ⎟⎥ ⎝ f1 ⎠ ⎦

−1

The choice of f1 was made on an experimental basis. It is found that this choice of f1

maintained the same peak amplitude mp with or without PE-DE. This satisfies the constraint of a fixed BT .

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Fig. 7.18: Bode plots of the response of PE-DE networks

The factor K is chosen such that the average power of the emphasized message signal is the same as that of the original message signal m ( t ) . That is,

K is such that W

∫

SM ( f ) d f =

−W

W

HPE ( f ) SM ( f ) d f = PM 2

∫

(7.40)

−W

This will ensure the same RMS bandwidth for the FM signal with or without PE.

(

Note that Brms

)FM

= 2 kf PM .

Example 7.8

and

Let SM ( f )

⎧ ⎪ ⎪ = ⎨1 + ⎪ ⎪ ⎩

HPE ( f ) =

⎡ K ⎢1 + ⎣

1 2

, f ≤ W

⎛f ⎞ ⎜ ⎟ ⎝ f1 ⎠ 0 , outside

⎛ f ⎞⎤ j ⎜ ⎟⎥ ⎝ f1 ⎠ ⎦

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Let us find (i) the value K and (ii) the improvement factor I , assuming f1 = 2.1 kHz and W = 15 kHz.

From Eq. 7.40, we have W

W

1

∫

⎛f ⎞ −W 1+ ⎜ ⎟ ⎝ f1 ⎠

or

K =

2

df =

∫

K df

−W

⎛W ⎞ f1 tan−1 ⎜ ⎟ W ⎝ f1 ⎠ W

K and

I =

∫

Cf2df

−W

⎡ 2 ⎢ C f 1+ ∫ ⎢ −W ⎣ W

2⎤

⎛f ⎞ ⎜ ⎟ ⎥ ⎝ f1 ⎠ ⎥⎦

(7.41a)

−1

df

where SN ( f ) is taken as C f 2 , C being a constant. Carrying out the integration, we find that ⎛W ⎞ tan− 1 ⎜ ⎟ ⎛W ⎞ ⎝ f1 ⎠ I = ⎜ ⎟ ⎝ f1 ⎠ 3 ⎡ W − tan− 1 ⎛ W ⎢ ⎜ ⎝ f1 ⎣ f1 2

(7.41b)

⎞⎤ ⎟⎥ ⎠⎦

With W = 15 kHz and f1 = 2.1 kHz, I ≈ 4 = 6 dB.

Pre-emphasis and de-emphasis also finds application in phonographic and tape recording systems. Another application is the SSB/FM transmission of telephone signals. In this, a number of voice channels are frequency division multiplexed using SSB signals; this composite signal frequency modulates the final carrier. (See Exercise 7.3.) PE-DE is used to ensure that each voice channel gives rise to almost the same signal-to-noise ratio at the destination.

7.45 Indian Institute of Technology Madras

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Example 7.9

Pre-emphasis - de-emphasis is used in a DSB-SC system. PSD of the message process is, SM ( f ) =

Let

1 ⎛f ⎞ 1+ ⎜ ⎟ ⎝ f1 ⎠

HPE ( f ) =

2

, f ≤ W

⎡ ⎛ f ⎞⎤ K ⎢1 + j ⎜ ⎟ ⎥ ⎝ f1 ⎠ ⎦ ⎣

where f1 is a known constant. Transmitted power with pre-emphasis remains the same as without preemphasis. Let us calculate the improvement factor I .

As the signal power with pre-emphasis remains unchanged, we have 2 ⎡ ⎛f ⎞ ⎤ ∫ SM ( f ) d f = ∫ SM ( f ) K ⎢⎢1 + ⎜⎝ f1 ⎟⎠ ⎥⎥ d f −W −W ⎣ ⎦ W

W

⎡ ⎢ ∞ ⎢ = K ∫ ⎢ − ∞⎢ ⎢1 + ⎣

⎤ ⎥ 1 ⎥ 2⎥ ⎛f ⎞ ⎥ ⎜ ⎟ ⎥ ⎝ f1 ⎠ ⎦

2 ⎡ ⎛f ⎞ ⎤ ⎢1 + ⎜ ⎟ ⎥ d f ⎢ ⎝ f1 ⎠ ⎥⎦ ⎣

That is, W

K =

SM ( f ) d f =

∫

−W

⎛W ⎞ f1 tan− 1 ⎜ ⎟ W ⎝ f1 ⎠

Noise power after de-emphasis, W

∫

Nout =

−W

=

N0 4

2 N0 HDE ( f ) d f 4

W

∫

−W

⎡ ⎢1 + ⎢ ⎣

⎛f ⎞ ⎜ ⎟ ⎝ f1 ⎠ K

2⎤

⎥ ⎥ ⎦

−1

df

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= W

N0 2

Note that the noise quantity at the output of the coherent demodulator is Noise power without de-emphasis = 2W ⋅

1 nc ( t ) . 2

N0 W N0 = 4 2

W N0 2 =1 Hence, I = W N0 2 This example indicates that PE-DE is of no use in the case of DSB-SC.

Exercise 7.2

A signal m ( t ) = 2 cos ⎡⎣(1000 π ) t ⎤⎦ is used to frequency modulate a very high frequency carrier. The frequency derivation produced is 2.5 kHz. At the output of the discriminator, there is bandpass filter with the passband in the frequency range 100 < f < 900 Hz. It is given that kd = 1 . a)

Is the system operating above threshold?

b)

If so, find the ( SNR )0 , dB.

Ans: (b) 34 dB

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Ac2 = 2 × 105 and 2 N0

Principles of Communication

Prof. V. Venkata Rao

Exercise 7.3

Consider the scheme shown in Fig. 7.19.

Fig. 7.19: Transmission scheme of Exercise 7.3

Each one of the USSB signals occupies a bandwidth of 4 kHz with respect to its carrier. All the message signals, mi ( t ) , i = 1, 2, ⋅ ⋅ ⋅, 10 , have the same power. m ( t ) frequency modulates a high frequency carrier. Let s ( t ) represent the FM signal. a)

Sketch the spectrum of s ( t ) . You can assume suitable shapes for M1 ( f ) , M2 ( f ) , ⋅ ⋅ ⋅, M10 ( f )

b)

At the receiver s ( t ) is demodulated to recover m ( t ) . (Note that from m ( t ) we can retrieve ⎡⎣ m j ( t ) ⎤⎦ , j = 1, 2, ⋅ ⋅ ⋅, 10 .) If m1 ( t ) can give rise to

signal-to-noise ratio of 50 dB, what is the expected signal-to-noise ratio from m10 ( t ) ?

7.48 Indian Institute of Technology Madras

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7.8 Noise Performance of a PCM system There are two sources of error in a PCM system: errors due to quantization and the errors caused by channel noise, often referred to as detection errors. We shall treat these two sources of error as independent noise sources and derive an expression for the signal-to-noise ratio expected at the output of a PCM system. As we have already studied the quantization noise, let us now look into the effects of channel noise on the output of a PCM system.

We will assume that the given PCM system uses polar signaling. Even if the transmitted pulse is rectangular, the received pulse pr ( t ) will be distorted due to a band-limited and imperfect channel; hence pr ( t ) may look like as shown in Fig. 7.20 (a).

Fig. 7.20: (a) Typical received pulse (without noise) (b) Received pulse with noise The input to the PCM receiver will be r ( t ) = ± pr ( t ) + n ( t ) , where n ( t ) is 2 a sample function of a zero-mean Gaussian process with variance σN . A

possible pulse shape of r ( t ) is shown in Fig. 7.20(b). The detection process of the PCM scheme consists of sampling r ( t ) once every Tb seconds and comparing it to a threshold. For the best performance, it would be necessary to sample pr ( t ) at its peak amplitude

( k ts

7.49 Indian Institute of Technology Madras

= k Tb + t p

)

resulting in a signal

Principles of Communication

Prof. V. Venkata Rao

component of ± Ap . Hence r ( k ts ) = ± Ap + N whose N is a zero mean Gaussian variable, representing the noise sample of a band-limited white Gaussian process with a bandwidth greater than or equal to the bandwidth of the PCM signal. If binary '1' corresponds to + Ap and '0' to − Ap , then,

( ) ( r '0' transmitted) is N ( − A , σ )

2 and fR ( r '1' transmitted ) is N Ap , σN

fR

p

2 N

where R (as a subscript) represents the received random variable. We assume that 1's and 0's are equally likely to be transmitted. The above conditional densities are shown in Fig. 7.21. As can be seen from the figure, the optimum decision threshold is zero. Let Pe,0 denote the probability of wrong decision, given that '0' is transmitted (area hatched in red); similarly Pe,1 (area hatched in blue). Then Pe , the probability of error is given by Pe =

1 1 Pe,0 + Pe,1 . 2 2

From Fig. 7.21, we have ∞

Pe,0 =

⎛ Ap ⎞ f r '0' d r Q = ( ) ⎜ ⎟ R ∫ ⎝ σN ⎠ 0

Fig. 7.21: Conditional PDFs at the detector input

⎛ Ap ⎞ Similarly Pe,1 = Q ⎜ ⎟ , which implies ⎝ σN ⎠

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⎛ Ap ⎞ Pe = Q ⎜ ⎟ ⎝ σN ⎠

(7.42)

For optimum results, the receiver uses a matched filter whose output is sampled once every Tb seconds, at the appropriate time instants so as to obtain the best possible signal-to-noise ratio at the filter output. In such a situation, it can be

⎛ Ap ⎞ shown that we can replace ⎜ ⎟ by ⎝ σN ⎠ received binary pulse and

2 Eb where Eb is the energy of the N0

N0 represents the spectral height of the, band-limited 2

⎛ Ap ⎞ white Gaussian noise process. Using the above value for ⎜ ⎟ yields, ⎝ σN ⎠ ⎛ 2 Eb Pe = Q ⎜ ⎜ N 0 ⎝

⎞ ⎟⎟ ⎠

(7.43)

Assuming that there are R binary pulses per sample and 2W samples/second, we have Tb =

1 where Tb represents the duration of each pulse. Hence ( 2 RW )

the received signal power Sr is given by, Sr =

Eb Sr = 2 RW Eb or Eb = . Tb 2 RW

Therefore Eq. 7.43 can also be written as

⎛ Sr Pe = Q ⎜ ⎜ RW N 0 ⎝

⎞ ⎟⎟ ⎠

⎛ γ ⎞ = Q⎜ ⎜ R ⎟⎟ ⎝ ⎠ where γ =

(7.44)

Sr . Eq. 7.42 to 7.44 specify the probability of any received bit W N0

being in error. In a PCM system, with R bits per sample, error in the reconstructed sample will depend on which of these R bits are in error. We would like to have an expression for the variance of the reconstruction error. Assume the following:

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i)

The quantizer used is a uniform quantizer

ii)

The quantizer output is coded according to natural binary code

iii)

Pe is small enough so that the probability of two or more errors in a block

of R bits is negligible.

Then, it can be shown that σc2 , the variance of the reconstruction error due to channel noise is, σc2

=

(

)

4 mp2 Pe 22 R − 1

( )

(7.45)

3 22 R

Details can be found in [3]. In addition to the reconstruction error due to channel 2 noise, PCM has the inevitable quantization noise with variance σQ =

∆ =

2 mp L

∆2 , where 12

and L = 2R . Treating these two error sources as independent noise

sources, total reconstruction noise variance σe2 , can be written as 2 σe2 = σQ + σc2

=

Let

mp2 3 L2

+

(

)

4 mp2 Pe L2 − 1

(7.46)

3 L2

M 2 ( t ) = PM

Then, ( SNR )0 =

=

PM σe2 ⎛P ⎞ ⎜ M⎟ 2 2 1 + 4 Pe L − 1 ⎜⎝ mp ⎟⎠ 3 L2

(

)

(7.47)

Using Eq. 7.44 in Eq. 7.47, we have

(SNR )0

=

3 L2 ⎛ 1 + 4 L − 1 Q ⎜⎜ ⎝

(

2

)

⎛P ⎞ ⎜ M⎟ ⎞ γ ⎜⎝ mp2 ⎟⎠ ⎟ R ⎟⎠

7.52 Indian Institute of Technology Madras

(7.48)

Principles of Communication

Prof. V. Venkata Rao

Figure 7.22 shows the plot of ( SNR )0 as a function of γ for tone modulation

⎛P 1⎞ ⎜ M2 = ⎟ . However, with suitable modifications, these curves are applicable ⎜ mp 2⎟ ⎝ ⎠ even in a more general case.

Fig. 7.22: ( SNR )0 performance of PCM Referring to the above figure, we find that when γ is too small, the channel noise introduces too many detection errors and as such reconstructed waveform has little resemblance to the transmitted waveform and we encounter 2 the threshold effect. When γ is sufficiently large, then Pe → 0 and σe2 ≈ σQ

which is a constant for a given n . Hence ( SNR )0 is essentially independent of γ resulting in saturation.

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For the saturation region, ( SNR )0 can be taken as

⎛P ⎞ = 3 22 R ⎜ M2 ⎟ ⎜ mp ⎟ ⎝ ⎠

( )

(SNR )0

(7.49)

The transmission bandwidth BPCM of the PCM system is k ( 2WR ) where k is a constant that is dependent on the signal format used. A few values of k are given below:

S. No.

Signal Format

k

1

NRZ polar

1

2

RZ polar

2

3

Bipolar (RZ or NRZ)

1

4

Duobinary (NRZ)

1 2

(For a discussion on duobinary signaling, refer Lathi [3]). As BPCM = k ( 2W R ) , we have R =

BPCM . Using this in the expression for ( SNR )0 in the saturation k ( 2W )

region, we obtain

(SNR )0

⎛P ⎞ = 3 ⎜ M2 ⎟ 2 ⎜ mp ⎟ ⎝ ⎠

BPCM kW

(7.50)

It is clear from Eq. 7.50 that in PCM, ( SNR )0 increases exponentially with the transmission bandwidth. Fig. 7.22 also depicts the ( SNR )0 performance of DSBSC and FM ( β = 2 , 5 ) . A comparison of the performance of PCM with that of FM is appropriate because both the schemes exchange the bandwidth for the signal -to -noise ratio and they both suffer from threshold phenomenon. In FM,

(SNR )0

increases as the square of the transmission bandwidth. Hence,

doubling the transmission bandwidth quadruples the output SNR . In the case of 7.54 Indian Institute of Technology Madras

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PCM, as can be seen from Eq. 7.49, increasing R by 1 quadruples the output 1 . As an R

SNR , where as the bandwidth requirement increases only by

example, if R is increased from 8 to 9, the additional bandwidth is 12.5% of that required for R = 8. Therefore, in PCM, the exchange of SNR for bandwidth is much more efficient than that in FM, especially for large values of R1. In addition, TP

PT

as mentioned in the introduction, PCM has other beneficial features such as use of regenerative repeaters, ease of mixing or multiplexing various types of signals etc. All these factors put together have made PCM a very important scheme for modern-day communications.

The PCM performance curves of Fig. 7.22 are based on Eq. 7.48 which is applicable to polar signaling. By evaluating Pe for other signaling techniques (such as bipolar, duobinary etc.) and using it in Eq. 7.47, we obtain the corresponding expressions for the

(SNR )0 .

It can be shown that the PCM

performance curves of Fig.7.19 are applicable to bipolar signaling if 3 dB is added to each value of γ .

Example 7.10

A PCM encoder produces ON-OFF rectangular pulses to represent ‘1’ and ‘0’ respectively at the rate of 1000 pulses/sec. These pulses amplitude modulate a carrier, Ac cos ( ωc t ) , where fc >> 1000 Hz. Assume that ‘1’s and ‘0’s are equally likely. Consider the receiver scheme shown in Fig. 7.23.

1 TP

PT

Note that the FM curve for β = 2 and PCM curve for R = 6 intersect at point A . The

corresponding γ is about 30 dB. If we increase improvement in

( SNR )0

γ beyond this value, there is no further

of the PCM system where as no saturation occurs in FM. If we take

BT = 2 ( β + 1) fm , then the transmission bandwidth requirements of PCM with R = 6 and FM

with β = 2 are the same. This argument can be extended to other values of R and β .

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Fig. 7.23: Receiver for the Example 7.11

⎧⎪2cos ( ωc t ) , if the input is '1' s (t ) = ⎨ 0 , if the input is '0' ⎪⎩ w ( t ) : sample function of a white Guassian noise process, with a two sided

spectral density of 0.25 × 10 − 4 Watts/Hz. BPF : Bandpass filter, centered at fc with a bandwidth of 2 kHz so that the signal is passed with negligible distortion. DD

a)

: Decision device (comparator)

Let Y denote the random variable at the sampler output. Find fY and fY

b)

1

0

( y 0)

( y 1) , and sketch them.

Assume that the DD implements the rule: if Y ≥ 1, then binary ‘1’ is transmitted, otherwise it is binary ‘0’.

If fY

1

( y 1)

can be well approximated by N ( 2, 0.1) , let us find the overall

probability of error. x ( t ) = s ( t ) + n ( t ) , where n ( t ) is the noise output of the BPF. = Ak cos ( ωc t ) + nc ( t ) cos ( ωc t ) − ns ( t ) sin ( ωc t )

⎧⎪2 , if the k th transmission bit is '1' where Ak = ⎨ th ⎪⎩0 , if the k transmission bit is '0' The random process Y ( t ) at the output of the ED, is Y ( t ) = R ( t ) cos [ ωc t + Θ ]

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Prof. V. Venkata Rao

{

R ( t ) = ⎡⎣ Ak + nc ( t ) ⎤⎦ + ⎡⎣ ns ( t ) ⎤⎦ 2

}

1 2 2

If ‘0’ is transmitted, Y ( t ) represents the envelope of narrowband noise. Hence, the random variable Y obtained by sampling Y ( t ) is Rayleigh distributed.

fY

0

( y 0)

=

(

y e N0

−

y2 2 N0

, y ≥ 0

)

where N0 = 0.25 × 10− 4 4 × 103 = 0.1 when ‘1’ is transmitted, the random variable Y is Rician, given by

fY

1

( y 1)

=

⎛ 2y ⎞ y I0 ⎜ ⎟ N0 ⎝ N0 ⎠

⎛ y2 + 4 ⎞ −⎜ ⎜ 2 N ⎟⎟ 0 ⎠ e ⎝

, y ≥ 0

These are sketched in Fig. 7.24.

Fig. 7.24: The conditional PDFs of Example 7.10

b) As the decision threshold is taken as ‘1’, we have ∞

Pe , 0 =

2

− 5y d y = e− 5 ∫ 10 y e 1 1

Pe , 1 =

∫

−∞

− 1 e 2 π 0.1

(y

− 2) 0.2

2

dy

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= Q

(

10

)

P ( error ) = Pe =

1 ⎡ −5 e + Q ⎡⎣ 10 ⎤⎦ ⎤ ⎦ 2⎣

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Appendix A7.1 PSD of Noise for Angle Modulated Signals For the case of strong predetection SNR , we have (Eq. 7.22b),

θ (t ) ϕ (t ) + λ (t ) =

Let

=

rn ( t ) sin ⎡⎣ψ ( t ) − ϕ ( t ) ⎤⎦ Ac

rn ( t ) sin ⎡⎣ψ ( t ) − ϕ ( t ) ⎤⎦ Ac 1 − Im ⎡⎢e ⎣ Ac

j ϕ( t )

rn ( t ) e

j ψ(t ) ⎤

⎦⎥

jϕ t jψ t Note that e ( ) is the complex envelope of the FM signal and rn ( t ) e ( ) is

nce ( t ) , the complex envelope of the narrow band noise.

Let xce ( t ) = e Then, λ ( t ) =

− j ϕ( t )

nce ( t ) = xc ( t ) + j xs ( t ) .

1 xs ( t ) Ac

We treat xce ( t ) to be a sample function of a WSS random process X ce ( t ) where, X ce ( t ) = e

− j Φ( t )

Nce ( t ) = X c ( t ) + j X s ( t )

(A7.1.1)

Similarly, Λ (t ) =

1 Xs (t ) Ac

(A7.1.2)

Eq. A7.1.2 implies, the ACF of Λ ( t ) , RΛ ( τ ) is

RΛ ( τ ) =

1 Ac2

R Xs ( τ )

(A1.7.3a)

S Xs ( f )

(A7.1.3b)

and the PSD

SΛ ( f ) =

1 Ac2

We will first show that

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E ⎡⎣ X ce ( t + τ ) X ce ( t ) ⎤⎦ = 0 E ⎡⎣ X ce ( t + τ ) X ce ( t ) ⎤⎦ = E ⎡Nce ( t + τ ) Nce ( t ) e ⎢⎣

− j ⎣⎡ Φ ( t + τ ) + Φ ( t ) ⎦⎤ ⎤

⎥⎦

(A7.4.1a) − = E ⎡⎣Nce ( t + τ ) Nce ( t ) ⎤⎦ E ⎡⎢e ⎣

j Φ(t + τ) + Φ( t ) ⎤

⎥⎦

(A7.4.1b) Eq. A7.4.1(b) is due to the condition that the signal and noise are statistically independent. E ⎡⎣Nce ( t + τ ) Nce ( t ) ⎤⎦ = ⎡⎣RNc ( τ ) − RNs ( τ ) ⎤⎦ + j ⎡⎣RNc Ns ( τ ) + RNs Nc ( τ ) ⎤⎦

As

RNc ( τ ) = RNs ( τ ) and RNc Ns ( τ ) = − RNs Nc ( τ ) ,

we have E ⎡⎣Nce ( t + τ ) Nce ( t ) ⎤⎦ = 0 Therefore E ⎡⎣ X ce ( t + τ ) X ce ( t ) ⎤⎦ = ⎡⎣R X c ( τ ) − R X s ( τ ) ⎤⎦ + j ⎡⎣R X c X s ( τ ) + R X s X c ( τ ) ⎤⎦ = 0

This implies R Xc ( τ ) = R X s ( τ ) . Now consider the autocorrelation of X ce ( t ) ,

{

∗ E ⎡ X ce ( t + τ ) X ce ( t )⎤⎦ = E ⎡⎢⎣e − ⎣

j Φ(t + τ)

jΦ t ∗ Nce ( t + τ ) ⎤⎥ ⎡⎢e ( ) Nce ( t )⎤⎥⎦ ⎦⎣

} (A7.1.5)

E ⎡ X ce ( ⎣

∗ ) X ce ( )⎤⎦

⎡ ⎤ ∗ = E ⎡Nce ( t + τ ) Nce ( t )⎤⎦ E ⎡⎢e j ⎣Φ( t ) − Φ(t + τ)⎦ ⎤⎥ ⎣ ⎣ ⎦

a1 + j b1

a2 + j b2

= ⎡⎣R X c ( τ ) + R X s ( τ ) ⎤⎦ + j ⎡⎣R X s X c ( τ ) − R X c X s ( τ ) ⎤⎦ = 2 R X s ( τ ) + j ⎡⎣R X s X c ( τ ) − R X c X s ( τ ) ⎤⎦

(Note that R X c ( τ ) = R X s ( τ ) as proved earlier.)

R Xs ( τ ) =

1 Re ⎡⎣( a1 + j b1 ) ( a2 + j b2 ) ⎤⎦ 2

7.60 Indian Institute of Technology Madras

(A7.1.6)

Principles of Communication

Prof. V. Venkata Rao

=

1 ( a1 a2 − b1 b2 ) 2

Now b1 = 2 RNs Nc ( τ ) = 0 for symmetric noise PSD, and a1 = 2 Rns ( τ ) . Hence,

R Xs ( τ ) =

{

}

1⎡ 2 RNs ( τ ) ⎤⎦ E cos ⎡⎣Φ ( t ) − Φ ( t + τ ) ⎤⎦ 2⎣

g ( τ)

That is,

S X s ( f ) = SNs ( f ) ∗ G ( f ) where g ( τ ) ←⎯→ G ( f ) . As

SΛ ( f ) = SΛ ( f ) =

1 Ac2

S Xs ( f ) , we have,

1 ⎡ SN ( f ) ∗ G ( f ) ⎤⎦ Ac2 ⎣ s

By definition, g ( τ ) is the real part of the ACF of e

− j Φ(t )

. We know that e

j Φ(t )

is

the complex envelope of the FM process. For a wideband PM or FM, the bandwidth of G ( f ) >> W , the signal bandwidth. The bandwidth of G ( f ) is approximately

BT , as shown in Fig. A7.1. 2

Fig. A7.1: Components of SΛ ( f ) : (a) typical G ( f )

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(b) SNs ( f )

Principles of Communication

Prof. V. Venkata Rao

Note that g ( 0 ) =

∞

∫ G (f ) d f

−∞

For

= E ⎡⎣cos ( 0 ) ⎤⎦ = 1 .

f ≤ W , G ( f ) ∗ SNs ( f ) N0

∞

∫ G (f ) d f

= N0 . That is, Φ ( t ) is

−∞

immaterial as far as PSD of Ns ( t ) in the range f ≤ W is concerned. Hence, we might as well set ϕ ( t ) = 0 in the Eq. 7.22 (b), for the purpose of calculating the noise PSD at the output of the discriminator.

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References 1)

Herbert Taub and D. L. Shilling, Principles of Communication systems, (2nd P

ed.) Mc Graw Hill, 1986 2)

Simon Haykin, Communication systems, (4th ed.) John Wiley, 2001

3)

B. P. Lathi, Modern digital and analog communication systems, Holt-

P

Saunders International ed., 1983

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P

P

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