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56 Chapter 2 DISCRETE-TIME SIGNALS AND SYSTEMS
1. Determine analytically the crosscorrelation r yx(l) interms of the autocorrelation r xx(l).
2. Let x(n) =cos(0.2πn)+0.5 cos(0.6πn), α =0.1, and k = 50. Generate 200 samples of
y(n) and determine its crosscorrelation. Can you obtain α and k by observing r yx(l)?
P2.11 Consider the following discrete-time systems:
T 1[x(n)] = x(n)u(n) T 2[x(n)] = x(n)+nx(n +1)
T 3[x(n)] = x(n)+ 1 2 x(n − 2) − 1 3 x(n − 3)x(2n) T4[x(n)] = ∑ n+5
k=−∞
T 5[x(n)] = x(2n)
T 6[x(n)] = round[x(n)]
where round[·] denotes rounding to the nearest integer.
1. Use (2.10) to determine analytically whether these systems are linear.
2. Let x 1(n) beauniformly distributed random sequence between [0, 1] over 0 ≤ n ≤ 100,
and let x 2(n) beaGaussian random sequence with mean 0 and variance 10 over
0 ≤ n ≤ 100. Using these sequences, verify the linearity of these systems. Choose any
values for constants a 1 and a 2 in (2.10). You should use several realizations of the above
sequences to arrive at your answers.
P2.12 Consider the discrete-time systems given in Problem P2.11.
1. Use (2.12) to determine analytically whether these systems are time-invariant.
2. Let x(n) beaGaussian random sequence with mean 0 and variance 10 over 0 ≤ n ≤ 100.
Using this sequence, verify the time invariance of the above systems. Choose any values
for sample shift k in (2.12). You should use several realizations of the above sequence to
arrive at your answers.
P2.13 For the systems given in Problem P2.11, determine analytically their stability and causality.
P2.14 The linear convolution defined in (2.14) has several properties:
x 1(n) ∗ x 2(n) =x 1(n) ∗ x 2(n)
: Commutation
[x 1(n) ∗ x 2(n)] ∗ x 3(n) =x 1(n) ∗ [x 2(n) ∗ x 3(n)] : Association
x 1(n) ∗ [x 2(n)+x 3(n)] = x 1(n) ∗ x 2(n)+x 1(n) ∗ x 3(n) :Distribution
x(n) ∗ δ(n − n 0)=x(n − n 0)
: Identity
(2.28)
1. Analytically prove these properties.
2. Using the following three sequences, verify the above properties.
x 1(n)=cos(πn/4)[u(n +5)− u(n − 25)]
x 2(n)=(0.9) −n [u(n) − u(n − 20)]
x 3(n)=round[5w(n)], −10 ≤ n ≤ 10; where w(n) isuniform over [−1, 1]
Use the conv m function.
P2.15 Determine analytically the convolution y(n) =x(n) ∗ h(n) ofthe following sequences, and
verify your answers using the conv_m function.
1. x(n) ={2, −4, 5, 3, −1, −2, 6}, h(n) ={1, −1, 1, −1, 1}
↑
↑
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