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Problems 55
1. Modify the evenodd function discussed in the text so that it accepts an arbitrary
sequence and decomposes it into its conjugate-symmetric and conjugate-antisymmetric
components by implementing (2.27).
2. Decompose the following sequence:
x(n) =10exp([−0.1+j0.2π]n), 0 ≤ n ≤ 10
into its conjugate-symmetric and conjugate-antisymmetric components. Plot their real
and imaginary parts to verify the decomposition. (Use the subplot function.)
P2.8 The operation of signal dilation (or decimation or down-sampling) isdefined by
y(n) =x(nM)
in which the sequence x(n) isdown-sampled by an integer factor M. For example, if
x(n) ={...,−2, 4, 3, −6, 5, −1, 8,...}
↑
then the down-sampled sequences by a factor 2 are given by
y(n) ={...,−2, 3, 5, 8,...}
↑
1. Develop a MATLAB function dnsample that has the form
function [y,m] = dnsample(x,n,M)
% Downsample sequence x(n) by a factor M to obtain y(m)
to implement the above operation. Use the indexing mechanism of MATLAB with
careful attention to the origin of the time axis n =0.
2. Generate x(n) =sin(0.125πn), − 50 ≤ n ≤ 50. Decimate x(n) byafactor of 4 to
generate y(n). Plot both x(n) and y(n) using subplot and comment on the results.
3. Repeat the above using x(n) =sin(0.5πn), − 50 ≤ n ≤ 50. Qualitatively discuss the
effect of down-sampling on signals.
P2.9 Using the conv_m function, determine the autocorrelation sequence r xx(l) and the
crosscorrelation sequence r xy(l) for the following sequences.
x(n) =(0.9) n , 0 ≤ n ≤ 20; y(n) =(0.8) −n , − 20 ≤ n ≤ 0
Describe your observations of these results.
P2.10 In a certain concert hall, echoes of the original audio signal x(n) are generated due to the
reflections at the walls and ceiling. The audio signal experienced by the listener y(n) isa
combination of x(n) and its echoes. Let
y(n) =x(n)+αx(n − k)
where k is the amount of delay in samples and α is its relative strength. We want to
estimate the delay using the correlation analysis.
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