Module 4. Signal Representation and Baseband Processing. Version 2 ECE IIT, Kharagpur

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1 Module Signal Representation and Baseband Processing Version ECE II, Kharagpur

2 Lesson 8 Response of Linear System to Random Processes Version ECE II, Kharagpur

3 After reading this lesson, you will learn about Modeling of thermal noise and power spectral density; ime domain analysis of a linear filter for random input; Representation of narrow-band Gaussian noise; Low-pass equivalent components of narrow-band noise; Band-pass Gaussian noise and its spectral density; A noise waveform is a sample function of a random process. hermal noise is epected to manifest in a communication receiver for an infinite time and hence theoretically noise may have infinite energy. hermal noise is typically modeled as a power signal. Usually, some statistical properties of thermal noise, such as its mean, variance, auto correlation function and power spectrum are of interest. hermal noise is further modeled as a wide-sense-stationary (WSS) stochastic process. hat is, if n(t) is a sample function of noise, a) the sample mean of n(t ), i.e. n(t) at t = t, is independent of the choice of sampling instant t and b) the correlation of two random samples, n(t ) and n(t ) depends only on the interval / delay (t -t ), i.e., Ent [ ( ) nt ( )] = Rn ( t t) = Rn ( τ ) he auto-correlation function (ACF), R (τ) of a WSS process, (t) is defined as: ACF = R ( τ ) = Ett [ ( ) ( + τ )]. R (τ) indicates the etent to which two random variables separated in time by τ vary with each other. Note that, R () = E[ ( t) ( t)] =, the mean square of (t). Power Spectral Density (psd) a. Specifies distribution of power of the random process over frequency f. If S (f) is the two-sided psd of (t), the power in a small frequency band Δf at f is [S (f ). Δf]; b. psd S (f) of thermal noise is a real, positive even function of frequency. he power in a band f to f is: f f f S ( f) df + S ( f) df = S ( f) df f f f ; in Volt /Hz For a deterministic waveform, the psd and ACF form a Fourier transform pair. he concept is etended to random processes and we may write for thermal noise process, [ ] S ( f) = F R ( τ ) = R ( τ ) e dτ.8. and R ( τ ) F [ S ( f) ] jϖτ = S( ) e j ϖτ = τ df Version ECE II, Kharagpur

4 Now, as noted in Lesson #7, the psd for white noise is constant: N Sn( f ) =.8. Hence, the ACF for such noise process is, N N Rn ( τ ) = F =. δ ( τ ).8.3 As we know, the signal, carrying information, occupies a specific frequency band and it is sufficient to consider the effect of noise, which manifests within this frequency band. So, it is useful to study the features of band-limited noise. For a base band additive white Gaussian noise (AWGN) channel of bandwidth W Hz, N Sn( f ) =, f < W =, Elsewhere.8. Simple calculation now shows that, the auto-correlation function for this base band noise is: R ( τ ) = WN sin c( Wτ ).8.5 n For pass-band thermal noise of bandwidth B around a centre frequency f c, the results can be etended: N B Sn( f ) =, f fc < =, Otherwise.8.6 he ACF now is given by, Rn ( τ ) = BN(sin cbτ).cosπ f c τ.8.7 In many situations, it is necessary to analyze the characteristics of a noise process at the output of a linear system, which transforms some ecitation given at its input. his is important because, the system being linear in nature, obeys the principle of superposition and if we ecite the system with a noise process and analyze the response noise process, we can use this knowledge for multiple situations. For eample, a specific case of interest may be to analyze the output of a linear filter when a noisy received signal is fed to it. For simplicity, we discuss about response of linear systems which are time-invariant. hough such analysis is more elegant when carried out in the frequency domain, we start with a time-domain analysis to provide some insight. ime-domain analysis for random input to a linear filter Let us consider a linear lowpass filter whose impulse response is h(t) and let us ecite the filter with white Gaussian noise. he input being a random process, it is not so important to get only an epression for the filter output y(t). It is statistically more significant to obtain epressions for the mean, variance, ACF and other parameters of the output signal. Now, in general, if (t) indicates the input to a linear system, the mean of the output y(t) Version ECE II, Kharagpur

5 is, y = h() t dt, where is the mean of the input process. he mean square value of the output is: y = R ( λ λ ) h ( λ ) h ( λ ) dλ dλ When the input is white noise, we know N Rn ( τ ) = δτ ( ), and y =.8.8 So, the mean square of the output noise process is: N y = δ ( λ λ) h( λ) h( λ) dλdλ N = h ( λ) d λ.8.9 E.8.: Let us consider a single-stage passive R-C lowpass filter whose impulse response is well known: ht () = e t ut (), where =. he 3 db cutoff frequency of the filter is f cutoff = R.C. It is straight forward to see that the average of noise at the output of this lowpass filter is zero: y = e dλ = n = πrc t Further, We observe that filter BW. N λ y = e dλ N N RC = π = N f cutoff = y, the noise power at the output of the filter, is proportional to the Auto-correlation function (ACF) In general, the autocorrelation of a random process at the output of a linear two-port network is: R ( τ ) = R ( λ λ τ) h( λ ). h( λ ) dλ dλ.8. y Specifically, for white noise, N R y ( τ ) = δλ ( λ τ) h( λ). h( λ) dλdλ Version ECE II, Kharagpur

6 N h ( λ). h ( λ τ) d λ = +.8. Considering the RC lowpass filter of E #.8., we see, N λ ( λ+ τ ) Ry ( τ ) = e e d λ N e τ =, τ As, R(τ) is an even function, R(τ) = R(- τ) and hence, N ( ) Ry τ = e τ.8. his is an eponentially decaying function of τ with a peak value of at τ =. As R y (τ) and the power spectrum S y (ω) are Fourier transform pair, we see that the power spectrum of the output noise is: N S y ( ω) =.8.3 ω + No Representation of Narrow-band Gaussian Noise Representation and analysis of narrow pass band noise is of fundamental importance in developing insight into various carrier-modulated digital modulation schemes which are discussed in Module #5. he following discussion is specifically relevant for narrowband digital transmission schemes. Let, (t) denote a zero-mean Gaussian noise process band-limited to ± B/ around centre frequency f c. here are several ways of analyzing such narrowband noise process and we choose an easy-to-visualize approach, which somewhat approimate. o start with, we consider a sample of a noise process over a finite time interval and apply Fourier series epansion while stretching the time interval to. If (t) is observed over an interval t, we may write π () t = ( cn cosnwt+ sn sin wt ) where, w = and and n= / = t ()cosnwtdt, n =,, cn / / = t ()sinnwtdt, n =,,.. sn /.8. Version ECE II, Kharagpur

7 It can be shown that cn and sn are Gaussian random variables. he centre frequency f c can now be brought in by the following substitution: nw = ( nw w ) + w c c t = cn nw wc t+ wct + sn nw wc t+ wc n= ( ) { cos[( ) ] sin[( ) t]} ()cos t w t ()sin t w t ;.8.5 c c s where, c() t = cncos( nw wc) t+ snsin( nw wc) t n= and () t = cos( nw w ) t+ sin( nw w ) t s cn c sn c n= c Another elegant and equivalent epression for (t) is: () t = c()cos t wct s()sin t wct = rt ( )cos[ wt+ Φ ( t)].8.8 where, c c s rt () = () t + () t ; s() t and Φ () t = tan ; Φ ( t) < π c () t It is easy to recognize that, () t = r()cos t Φ () t and () t = r()sin t Φ () t c Low pass equivalent components of narrow band noise Let, ct and st represent samples of c (t) and s (t). hese are Gaussian distributed random variables with zero mean as the original noise process has zero mean. E = E =.8.9 [ ] [ ] ct st Now, an epression for variance of ct, by definition, looks like the following: [ cn cos( nw wc ) t + sn sin( nw wc ) t] E ct = E n= m= [ cm cos( mw wc ) t + sm sin( mw wc ) t].8. However, after some manipulation, the above epression can be put in the following form: E ct [ cn. cm cos( nw wc ) t.cos( mw wc ) t+ cn. sm cos( nw wc ) t. = sin( mw wc) t sn. cmsin( nw wc) tco. s( mw wc) t n= m= + sn. sm sin( nw wc ) t.sin( mw wc ) t].8. s Version ECE II, Kharagpur

8 Here, / / cncm = E ().cos t nwtdt. ().cos t mwtdt / / / / ( t ) ( t)cos nwt.cosmwtdtdt / / / / R( t t)cos nwt.cos mwtdtdt.8. / / = = R (t -t ) in the above epression is the auto-correlation of the noise process (t). Now, putting t t = u and t v =, we get, ( + v) / cncm = cos nwv R( u).cos mw( u v. ) du dv + / ( + v) ( + v) / π mu = cos nv cos mv R ( u).cos du π π / ( + v) ( + v) π mu sin π mv R ( u).sin du dv ( + v).8.3 m Now for and putting f m =, where f m tends to for all m, the inner integrands are: and ( + v) lim π m = ( + v) ( + v) lim R( u).sinπ fmudu = ( + v) R ( u).cos f udu S ( fm ), say m Let us choose to write, lim fm = = f Now from Eq..8.3, we get a cleaner epression in the limit: Version ECE II, Kharagpur

9 lim cn cm / = S ( f) cos πnvcos πmvdv / = S( f), m = n =, m n.8. Following similar procedure as outlined above, it can be shown that, lim sn sm = S ( f), m = n =, m n.8.5 and =, all m, n.8.6 lim cn sm Eq establish that the coefficients c -s and s -s are uncorrelated as approaches. Now, referring back to Eq..8., we can see that, E ct = cn cos ( nw wc ) t + sin ( nw wc ) t.8.7 Here = lim n= n= S ( f ) = lim m S ( f) df t denotes the mean square value of (t). = t.8.8 Similarly, it can be shown that, E [ st ] = E[ ct ] = t.8.9 Since, = =, we finally get, σ = σ = σ, the variance of (t). st ct st ct It may also be shown that the covariance of ct and st approach as approaches. herefore, ultimately it can be shown that ct and st are statistically independent. So, ct and st are uncorrelated Gaussian distributed random variables and they are statistically independent. hey have zero mean and a variance equal to the variance of the original bandpass noise process. his is an important observation. ct is called the in-phase component and st is called the quadrature component of the noise process. Spectral Density of In-phase and Quadrature Component of Bandpass Gaussian Noise Following similar procedures as adopted for determining mean square values of c (t) and s (t), we can compute their auto-correlation and cross-correlation functions as below: Version ECE II, Kharagpur

10 ACF of = R ( τ ) = S ( f). cos π ( f f ) τdf.8.3 Rs ( ) ct c τ = ACF of st c c = S ( f). cos π ( f f ) τ df = ( τ ).8.3 Cross Correlation Function (CCF) between ct and st is: R ( τ ) = S ( f).sin π( f f ) τdf.8.3 c s c and R ( τ ) = R ( τ ).8.33 c s c s Eq..8.3 can be epressed in the following convenient manner: R ( τ) = S ( f)cos π( f f ) τdf + S ( f)cos π( f f ) τ( df) R c c c c fc τ = S ( f + f )cos π fτdf + S ( f f )cos π f df fc fc c c [ ] = S( f + fc) + S( f fc) cosπ fτdf.8.3 fc From this, using inverse Fourier ransform, one gets, S ( f) = S ( f + f ) + S ( f f c ).8.35 c c Moreover, S ( f) = S ( f) = S ( f + f ) + S ( f f ).8.36 c s c c Note that the power spectral density of c (t) is the sum of the negative and positive frequency components of S (f) after their translation to the origin. he following steps summarize the method to construct psd of S c ( f ) or S s ( f) from S (f): a. Displace the +ve frequency portion of the plots of S (f) to the left by f c. b. Displace the ve frequency portion of S (f) to the right by f c. c. Add the two displaced plots. If f c is not the centre frequency, the psd of c (t) or s (t) may be significantly different from what may be guessed intuitively. Problems Q.8.) Consider a pass band thermal noise of bandwidth MHz around a center frequency of 9 MHz. Sketch the auto co-relation function of this pass band thermal noise normalized to its PSD. Q.8.) Sketch the pdf of typical narrow band thermal noise. Version ECE II, Kharagpur