第四章习题4.1 (a) By expanding the equation()()[]()⎥⎦⎤⎢⎣⎡==⎰--∞→∞→2200021T 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 that2102211)(1)(∑∑-=---+-==N n fn j fm j N N m xx en x N e m r ππ.4.2 For zero-mean, jointly Gaussian random variables, X 1, X 2, X 3, X 4, itis 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*22varwhich is defined as[][][]22(()(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 Gaussianrandom 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 noiseprocess 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 asignal 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 thatthe spectral window is narrower than the peak.(b) Explain why it is not advantageous to increase M beyond thevalue obtained in part (a).4.7 The N-point DFT of a random sequence x(n) is ∑-=-=10/2)()(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 withvariance 2ωσ.(a) Determine the parameters of the MA(2), MA(4), and MA(8)models that provide a minimum mean-sequare error fit to thedata x(n).(b) Plot the true spectrum and those of the MA (q), q=2,4,8spectra and compare the results. Comment on how well theMA(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 withvariance 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, andcompare the results. Comment on how well the AR(p) modelsappoximate the MA (2) process.4.10 The autocorrelation sequence for an AR process x(n) ismxx m ⎪⎭⎫ ⎝⎛=41)(γ (a) Determine the difference equation for x(n)(b) Is your answer unique? If not, give any other possiblesolutions.4.11 Suppose that we represent an ARMA(p,q) process as a cascade ofan MA(q) followed by an AR(p) model. The input-output equation for the MA(q) model is ∑=-=qk k k n w b n v 0)()(, where w(n) is a whitenoise process. The input-output equation for the AR(p) model is∑==-+pk k n v k n x a n x 1)()()((a) By computing the autocorrelation of v(n), show thatq m d b b m mq k m w m k k w vv ≤≤==∑-=+0)(022σσγ(b) Show that 1)()(00=+=∑=a k m a m pk vx k vv γγ4.12 Suppose that the AR(2) process in Problem 4.8 is corrupted by anadditive white noise process v(n) with variance 2v σ. Thus, we havey(n)=x(n)+v(n)(a) Determine the difference equation for y(n) and thusdemonstrate that y(n) is an ARMA(2,2) process. Determinethe coefficients of the ARMA process.(b) Generalize the result in part (a) to an AR(p) process∑=+--=pk 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 maybe expressed as the solution to the equationa 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 withrandom 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 σthe 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 afunction of 2w σ and 1f .(b) Determine the reflection coefficients 1K and 2K correspondingto 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 signalin 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 positiveinteger, is unknown, and is to be determined by cross-correlating x(n) with y(n). Assume that 0n M N +>. Let∑-=-=10)()()(N n xy n x m n y m rdenote 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 toerrors in determining the unknown delay.(a) For 0n m =, determine ()[]0n r E xy . Also, determine thevariance ()[]0var n r xy , due to the presence of the noise. In bothcalculations, assume that the double-frequency term averages to zero. That is, 0/2ωπ>>M .(b) Determine the signal-to-noise ratio, defined as []{}[])(var )(020n r n r E SNR xy xy = (c) What is the effect of the pulse duration M on the SNR?。