Computer Exercises in Adaptive Filters
Computer Exercises in Adaptive Filters
Computer Exercises in Adaptive Filters
Computer Exercises
Generate a stationary process AR(2) denoted by s n . Suppose that
s n a1 s n 1 a2 s n 2 n
with a variable SNR. Put the received the signal to a Wiener-filter, the length of
the filter is N. And the output is denoted by y n .
Study on the relationship between the cost function and SNR of the signal,
provided the length of the Wiener-filter is given.
Study on the relationship between the cost function and length of the filter,
provided the SNR is given.
3. When one-step prediction is done, how about the cost function varies with the
SNR and length of the Wiener-filter.
where the sample v n is drawn from a white-noise process of zero mean and
variance v2 . The AR parameters a1 and a2 are chosen so that the roots of the
characteristic equation
1 a1 z 1 a2 z 2 0
are complex; that is, a12 4a2 . The particular values assigned to a1 and a2 are
determined by the desired eigenvalue spread R . For specified values of a1
and a2 , the variance of v2 of the white-noise v n is chosen to make the
process x n have variance x2 1 .
3. By adaptive filtering, do the research work on the picking up of a signal with only
one frequency.
Suppose that x(t ) s t cos(2ft ) ,here s t is a wideband signal, f is
arbitrary chosen, you are required to extract cos 2ft .
Suppose that
s t
is a
f , N .
Here N is the
where
Given different sets of AR parameters, fixed the step-size parameter, and the
(0 ) 0 .
initial condition w
( n) including the single
Please plot the transient behavior of weight w
5. Study the use of the LMS algorithm for adaptive equalization of a linear dispersive
channel that produces (unknown) distortion. Assume that the data are all real
valued. Fig. P5 shows the block diagram of the system used to carry out the study.
Random-number generator 1 provides the test signal x n , used for probing the
channel, whereas random-number generator 2 serves as the source of additive
white noise v n that corrupts the channel output. These two random-number
generators are independent of each other. The adaptive equalizer has the task of
correcting for the distortion produced by the channel in the presence of the
additive noise. Random-number generator 1, after suitable delay, also supplies the
desired response applied to the adaptive equalizer in the form of the training
sequence.
The experiment is required
To evaluate the response of the adaptive equalizer using the LMS algorithm
to changes in the eigenvalue spread (R ) and step-size parameter .
The effect on the squared error when length of the delay and the length of the
data is changed.
6. Generate 100 samples of a zero-mean white noise sequence (n) with variance
1
, by using a uniform random number generator.
12
(a) Compute the autocorrelation of (n) for 0 m 15 .
rk (m) , 1 k 10
and 0 m 15 .
1 10
rk (m) and
10 k 1
1 az
1
0.99 z 1 az 1 0.98 z 2
2
(a) Sketch a typical plot of the theoretical power spectrum xx ( f ) for a small
value of the parameter a (i.e., 0<a<0.1). Pay careful attention to the value of the
two spectral peaks and the value of Pxx ( ) for / 2 .
(b) Let a=0.1. Determine the section length M required to resolve the spectral
peaks of xx ( f ) when using Bartletts method.
(c) Consider the Blackman-Turkey method of smoothing the periodogram. How
many lags of the correlation estimate must be used to obtain resolution
comparable to that of the Bartlett estimate considered in part (b)? How many
data must be used if the variance of the estimate is to be comparable to that of a
four-section Bartlett estimate?
(d) Generate a data sequence x(n) by passing white Gaussian noise through H(z)
and compute the spectral estimates based on the Bartlett and Blackman-Tukey
methods and, thus, confirm the results obtained in parts (b) and (c).
(e) For a=0.05, fit an AR(4) model to 100 samples of the data based on the YuleWalker method, and plot the power spectrum. Avoid transient effects by
discarding the first 200 samples of the data.
Fig. P9
(a) Determine the quadratic performance index and the optimum parameters for the
signal
x n sin
n
n
4
1
max .
10
(c) Repeat the experiment in part (b) for N 10 trials with different noise sequences,
and compute the average values of the predictor coefficients. Comment on how
these results compare with the theoretical values in part (a).
iteration number.
(b) Repeat part (a) for 10 trials, using different noise sequences, and superimpose
the 10 plots of a1 n and a2 n .
(c) Plot the learning curve for the average (over the 10 trials) MSE for the data in
part (b).
11. A random process x n is given as
x n s n n sin 0 n n ,
0 , 0
4
x n a1 x n 1 a2 x n 2 v n
f n x n x n
1 n a1 w 1 n
and
2 n a2 w 2 n
Using power spectral plots of
f n, 1 n
13. Consider a linear communication channel whose transfer function may take one of
Gaussian noise with zero mean and variance v2 0.01 . The channel input xn
where the dashed lines are included to identify the submatrices that
correspond to vary filter lengths.
where the value of the fourth entry ensures that the model parameter a3 is
zero.
3.
4.