数字信号处理第四章习题

第四章习题

4.1 (a) By expanding the equation

()()[]()??????==?--∞→∞

→2

200021T T Ft j T xx T xx dt e t x T E lim F P E lim F 00πΓ taking the expected value, and finally taking the limit as ∞→0T ,

show that the right-hand side converges to )(f xx Γ.

(b) Prove that

2102211)(1)(∑∑-=---+-==N n fn j fm j N N m xx e

n x N e m r ππ.

4.2 For zero-mean, jointly Gaussian random variables, X 1, X 2, X 3, X 4, it

is well known that

)()()()()()()(3241423143214321X X E X X E X X E X X E X X E X X E X X X X E ++=. Use this result to derive the mean-square value of ()m r xx and the variance, given by

()[][]()()()[]∑∞-∞=+-+-≈n xx xx xx xx m n m n n m N N m r γγγ*22

var

which is defined as

[][][]2

2(()(var m r E m r E m r xx xx xx -=. 4.3 By use of the expression for the fourth joint moment for Gaussian

random variables, show that

(a)()()[]??

??????????????--+??????+++=2212122121421)(sin )(sin )(sin )(sin 1f f N N f f f f N N f f f P f P E x xx xx ππππσ (b)[]??

??????????????--+??????++=2212122121421)(sin )(sin )(sin )(sin )()(cov f f N N f f f f N N f f f P f P x xx xx ππππσ

(c)[]??

???????????? ??+=242sin 2sin 1)(var f N fN f P x xx ππσ under the condition that the sequence ()n x is a zero-mean white Gaussian noise sequence with variance 2x σ.

4.4 Generalize the results in Problem 4.3 to a zero-mean Gaussian noise

process with power density spectrum )(f xx Γ, as given by

()[]()???

????????? ??+Γ=222sin 2sin 1var f N fN f f P xx xx ππ (Hint: Assume that the colored Gaussian noise process is the output of a linear system excited by white Gaussian noise.)

4.5 Show that the periodogram values at frequencies

,1,1,0,/-==L k L k f k given by (4.1.35), can be computed by passing the sequence through a bank of L IIR filters, where each filter has an impulse response )()(/2n u e n h N nk j k π-= and then computing the magnitude-squared value of the filter outputs at n=N. Note that each filter has a pole on the unit circle at the frequency f k .

4.6 The Bartlett method is used to estimate the power spectrum of a

signal x(n). We know that the power spectrum consists of a single peak with a 3 dB bandwidth of 0.01 cycle per sample, but we do not know the location of the peak.

(a) Assuming that N is large, determine the value of M=N/K so that

the spectral window is narrower than the peak.

(b) Explain why it is not advantageous to increase M beyond the

value obtained in part (a).

4.7 The N-point DFT of a random sequence x(n) is ∑-=-=1

0/2)()(N n N nk j e n x k X π.

Assume that E[x(n)]=0 and E[x(n)x(n+m)]=)(2m w δσ (in other words,

x(n) is a white noise process).

(a) Determine the variance of X(k).

(b) Determine the autocorrelation of X(k).

4.8 An AR(2) process is described by the difference equation

)()2(81.0)(n n x n x ω+-=, where w(n) is a white noise process with

variance 2ωσ.

(a) Determine the parameters of the MA(2), MA(4), and MA(8)

models that provide a minimum mean-sequare error fit to the

data x(n).

(b) Plot the true spectrum and those of the MA (q), q=2,4,8

spectra and compare the results. Comment on how well the

MA(q) models approximate the AR (2) process.

4.9 An MA (2) process is described by the difference equation )2(81.0)()(-+=n n n x ωω, where w(n) is a white noise process with

variance 2ωσ.

(a) Determine the parameters of the AR(2), AR(4), and AR(8)

models that provide a minimum mean-square error fit to the data x(n).

(b) Plot the true spectrum and those of the AR(p), p=2,4,8, and

compare the results. Comment on how well the AR(p) models

appoximate the MA (2) process.

4.10 The autocorrelation sequence for an AR process x(n) is

m

xx m ??? ??=41)(γ (a) Determine the difference equation for x(n)

(b) Is your answer unique? If not, give any other possible

solutions.

4.11 Suppose that we represent an ARMA(p,q) process as a cascade of

an MA(q) followed by an AR(p) model. The input-output equation for the MA(q) model is ∑=-=q

k k k n w b n v 0)()(, where w(n) is a white

noise process. The input-output equation for the AR(p) model is

∑==-+p

k k n v k n x a n x 1)()()(

(a) By computing the autocorrelation of v(n), show that

q m d b b m m

q k m w m k k w vv ≤≤==∑-=+0)(022

σσγ

(b) Show that 1)()(00=+=∑=a k m a m p

k vx k vv γγ

4.12 Suppose that the AR(2) process in Problem 4.8 is corrupted by an

additive white noise process v(n) with variance 2v σ. Thus, we have

y(n)=x(n)+v(n)

(a) Determine the difference equation for y(n) and thus

demonstrate that y(n) is an ARMA(2,2) process. Determine

the coefficients of the ARMA process.

(b) Generalize the result in part (a) to an AR(p) process

∑=+--=p

k k n w k n x a n x 1)()()( and )()()(n v n x n y +=.

4.13 The harmonic decomposition problem considered by Pisarenko may

be expressed as the solution to the equation

a a a Γa H w yy H 2σ=

The solution for a may be obtained by minimizing the quadratic form a Γa yy H subject to the constraint that a a H =1. The constraint can be incorporated into the performance index by means of a Lagrange multiplier. Thus the performance index becomes

()a a a Γa H yy H 1-+=λζ.

By minimizing ζ with respect to a , show that this formulation is equivalent to the Pisarenko eigenvalue problem given in (4.4.9), with the Lagrange multiplier playing the role of the eigenvalue. Thus,

show that the minimum of ζ is the minimum eigenvalue 2w σ.

4.14 The autocorrelation of a sequence consisting of a sinusoid with

random phase in noise is

)(2cos )(21m m f P m w xx δσπγ+=

where 1f is the frequency of the sinusoidal, P its power, and 2w σ

the variance of the noise. Suppose that we attempt to fit an AR(2) model to the data.

(a) Determine the optimum coefficients of the AR(2) model as a

function of 2w σ and 1f .

(b) Determine the reflection coefficients 1K and 2K corresponding

to the AR(2) model parameters.

(c) Determine the limiting values of the AR(2) parameters and (1K ,2K )

as 02→w σ.

4.15 This problem involves the use of cross-correlation to detect a signal

in noise and estimate the time delay in the signal. A signal x(n) consists of a pulsed sinusoid corrupted by a stationary zero-mean white noise sequence. That is, 10),()()(0-≤≤+-=N n n w n n y n x ,

where )(n w is the noise with variance 2w σ and the signal is

???-≤≤=otherwise M n n A n y ,

010,cos )(0ω. The frequency 0ω is known, but the delay 0n , which is a positive

integer, is unknown, and is to be determined by cross-correlating x(n) with y(n). Assume that 0n M N +>. Let

∑-=-=1

0)()()(N n xy n x m n y m r

denote the cross-correlation sequence between x(n) and y(n). In the absence of noise, this function exhibits a peak at delay 0n m =. Thus,

0n is determined with no error. The presence of noise can lead to

errors in determining the unknown delay.

(a) For 0n m =, determine ()[]0n r E xy . Also, determine the

variance ()[]0var n r xy , due to the presence of the noise. In both

calculations, assume that the double-frequency term averages to zero. That is, 0/2ωπ>>M .

(b) Determine the signal-to-noise ratio, defined as []{}[])(var )(0

2

0n r n r E SNR xy xy = (c) What is the effect of the pulse duration M on the SNR?