Principles Of Adaptive Filters And Self-learning Systems 1st Edition Anthony Zaknich - Solutions

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7.8. Compute the MV spectral estimate of a random phase complex exponential process in white noise, defined by,[ ][ ] has a variance of .is a random variable uniformly distributed over , .where :[ ] [ ]2 11 wj jn w n x n A e e w nσφ π πφ θ−= +Use Woodbury’s identity to find the required
7.7. Compute the optimum filter for estimating the Minimum Variance (MV)power spectrum of white noise having a variance of 2σ x .
7.6. By looking at the performance comparisons of the various nonparametric spectral estimation methods what general conclusions can be drawn?
7.5. For a total data length N what can you say about the relationship between the resolution and variance as function of K, the number of nonoverlapping data sections, for Bartlett’s spectral estimation method?
7.4. Does the variance of the periodogram spectral estimate of white noise reduce as the data length N increases?
7.3. Assume that a random process can be described by two equal amplitude sinusoids in unit random variance white noise as defined by the following equation, x[n] = Asin(nθ1 +φ1) + Asin(nθ 2 +φ 2 ) + v[n]Also assume that Δθ = θ1 −θ 2 = 0.02π radians and Fs = 1000 Hz. What is the minimum
7.2. What is the power spectrum of white noise having a variance of σ x 2 ?
7.1. What are the two main approaches to spectral estimation and in what way do they differ?
6.4. Can the exponentially weighted RLS algorithm (refer to Chapter 8) be considered to be a special case of the Kalman filter?
6.3. Use a Kalman filter to estimate the first-order AR process defined by, x[n] = 0.5x[n −1]+ w[n] , where w[n] is zero mean white noise with a varianceσ w2 = 0.64 .The noisy measurements of x[n] are defined by the equation, y[n] = x[n]+ v[n] ,where v[n] is unit variance zero mean white noise
6.2. Develop a Kalman filter to estimate the value of an unknown scalar constant x given measurements that are corrupted by an uncorrelated, zero mean white noise v[n] that has a variance of σ v 2 .
7.9. Compute the qth order Maximum Entropy (ME) spectral estimate for Problem 7.8.
8.2. In Problem 8.1 if the autocorrelation sequence rx (k) is given as rx (0) = 5.7523 and rx (1) = 4.0450 find the maximum bound for the step size in the LMS algorithm.
8.1. Assume x[n] is a second-order autoregressive process defined by the difference equation, x[n] = 1.2728x[n −1] − 0.81x[n − 2]+ v[n] , where v[n] is unit variance white noise the optimum causal predictor for x[n] is given by, xˆ[n] = 1.2728x[n −1]− 0.81x[n − 2] .Show how you could
9.6 What is the disadvantage of the circular convolution method?
9.5. What is the main purpose of the so called overlap-save and the overlapadd methods in frequency domain filtering?
9.4. For frequency domain filtering define the inverse of the p x p DFT matrix F.
9.3. What computational savings can be had by using a radix-2 decimation-intime FFT to perform the DFTs required in Problem 9.2?
9.2. Show how linear convolutions on sequences can be done by using frequency domain operations.
9.1. Compute how many multiplication operations are saved in the coefficient update equation for a time domain BLMS algorithm per block of L input points. Is it worth the trouble?
8.6. For the exponentially weighted RLS algorithm prove thatΦ(n)w[n] =θ (n) by setting the derivative of the weighted error ξ(n) with respect to w*[n] to zero, given that,????=Φ = −n kn n k k T k 0* ] [ ] [ ) ( x x λ and, ????== −n kn n kd k k 0θ ( ) λ [ ]x*[ ]
8.5. Compute and sketch a graph of the expectation of the squared error as a function of the single real-valued filter coefficient for a simple zero-order MA(0) process .
8.4. If the process and the FIR filter coefficients are complex valued show that the adaptive LMS update equation is w[k+1] = w[k] +μ e[k] x*[k].
8.3. Use the normalised LMS algorithm to derive the FIR filter coefficient update equations for the second-order AR(2) linear prediction equation of Problem 8.1.
6.1. Is the Kalman filter useful for filtering nonstationary and nonlinear processes?
2.2. Show how the z-Transform can become the DFT.
2.12. Given the autocorrelation function for the random phase sinusoid in the previous Problem 2.11 compute the 2 x 2 autocorrelation matrix.v2.13. The autocorrelation sequence of a zero mean white noise process is rv (k) =σ v2δ (k) and the power spectrum is ( ) v2 jPv e σ θ = , where 2σ v is
2.11. What is the mean and autocorrelation of the random phase sinusoid defined by, ) sin( ] [ 0 φ ω + = n A n x , given that A and ω0 are fixed constants and φ is a random variable that is uniformly distributed over the interval−π toπ . The probability density function for φ is,Principles
2.10. Compute the rounding quantization error variance for an Analogue to Digital Converter (ADC) with a quantization interval equal to Δ. Assume that the signal distribution is uniform and that the noise is stationary white noise.
2.9. Find the eigenvalues of the following 2 x 2 Toeplitz matrix, A =???????? ???????????? ????a b b a Find the eigenvectors for a = 4 and b = 1.
2.8. Why are ergodic processes important?
2.7. Which of the following matrices are orthogonal? Compute the matrix inverses of those that are orthogonal.a.0 1 0 0 0 1 1 0 0????????????????????????????????????????, b.0 0 1 0 1 0 1 0 0????????????????????????????????????????, c.1 1 1 1 1 1 1 1 1????????????????????????????????????????, d.1 0
2.6. Which of the following matrices are Toeplitz?a.3 2 1 4 3 2 5 4 3????????????????????????????????????????, b.1 2 3 1 2 3 1 2 3????????????????????????????????????????, c.1 1 1 1 1 1 1 1 1????????????????????????????????????????, d.3 2 1 2 1 2 1 2 3????????????????????????????????????????, e.1 2
2.5. Which of the following vector pairs are orthogonal or orthonormal?a. [1, -3, 5]T and [-1, -2, -1]Tb. [0.6, 0.8]T and [4, -3]Tc. [0.8, 0.6]T and [0.6, -0.8]Td. [1, 2, 3]T and [4, 5, 6]T
2.4. Given the following FIR filter impulse responses what are their H(z) and H(z-1)? What are the zeros of the filters? Express the transfer functions in terms of zeros and poles? Prove that these filters have linear phases.a. h[n] = {0.5, 0.5}b. h[n] = {0.5, 0.0, -0.5}c. h[n] = {0.5, 0.0, 0.5}d.
2.3. Which of the FIR filters defined by the following impulse responses are linear phase filters?a. h[n] = {0.2, 0.3, 0.3, 0.2}b. h[n] = {0.1, 0.2, 0.2, 0.1, 0.2, 0.2}c. h[n] = {0.2, 0.2, 0.1, 0.1, 0.2, 0.2}d. h[n] = {0.05, 0.15, 0.3, -0.15,-0.15}e. h[n] = {0.05, 0.3, 0.0, -0.3, -0.05}
2.1. Does the equation of a straight line y = α x + β, where α and β are constants, represent a linear system? Show the proof.
2.15. If x[n] is a zero mean wide-sense stationary white noise process and y[n] is formed by filtering x[n] with a stable LSI filter h[n] then is it true that, and are the variances of [ ] and [ ] respectively.
5.3 as σ v2 →0 .
5.8. Find the optimum causal and noncausal IIR Wiener filters for estimating a zero-mean signal s[n] from a noisy real zero-mean signal x[n] = s[n]+ v[n] , where v[n] is a unit variance white noise zero-mean process uncorrelated with s[n] and s[n] is a first-order AR process defined by, s[n] =
5.7. Compute the minimum mean square error for Problem 5.6 for 1 to 5 step predictors. What do you find odd about the sequence of errors from step 1 through to step 5?
5.6. Consider a random process whose autocorrelation sequence is defined by,r (k) (k) (0.9) cos( k / 4) k x =δ + π . The first six autocorrelation values are, rx [ ]T = 2.0 0.6364 0 −0.5155 −0.6561 −0.4175 .Solve the equations for a first-order three step linear predictor and compute the
5.5. Show that the solution to Problem 5.4 approaches the solution to Problem
5.4. Reconsider Problem 5.3 when the measurement of x[n] is contaminated with zero-mean white noise having a variance of 2σ v , i.e., y[n] = x[n]+ v[n] . Find the optimum first-order linear predictor in this case.
5.3. Find the optimum first-order linear predictor having the form xˆ[n +1] = w[0]x[n]+ w[1]x[n −1] , for a first-order AR process x[n] that has an autocorrelation sequence defined by rs k( ) =α k .
5.2. In Problem 5.1 the optimum first-order FIR Wiener filter for σ v2 =1 andα = 0.8 is, W(z) = 0.4048 + 0.2381z−1 . What is the signal to noise ratio(S/N) improvement, computed in dB, achieved by the Wiener filtering?
5.1. Find the optimum first-order FIR Wiener filter for estimating a signal s[n]from a noisy real signal x[n] = s[n]+ v[n] , where v[n] is a white noise process with a variance 2σ v that is uncorrelated with s[n]. Assume that s[n] is a first-order AR process having an autocorrelation sequence
3.4. In the Acoustic Positioning System example why were the modified set of model equations, G0 H0 Tused instead of the original model equations, Tcc Tc1 Tc3 Tc2 Tc0 to solve the LSE problem?
3.3. Assume that you have three independent measurements, y1, y2 and y3 with a measurement error variance of σ 2, of the volume of water in the tray of optimum volume (tray of Problem 3.1). Show how you would use LSE to solve for x, the tray depth.
3.1. Assume that you have a square piece of sheet steel which you wish to bend up into a square open tray. The sheet is 6 by 6 units in area and the bend lines are x units in from the edge as shown in the diagram below,Solve for x to make a tray having the maximum volume, using differential
3.2. Show how you could use the LSE equations to solve Problem 3.1.

Principles Of Adaptive Filters And Self-learning Systems 1st Edition Anthony Zaknich - Solutions

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