Discrete Time Signal and System Assignment

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Problem 2.1 Determine whether or not each of the following sequences is periodic.

If your answer is yes, determine the period.

Problem 2.2 A sequence x(n) is shown below. Express x(n) as a linear combination of weighted and delayed unit samples.

Figure P2.2-1 Matlab Assignment Experts

Problem 2.3 For each of the following systems, y(n) denotes the output and x(n) the input. Determine for each whether the specified input-output relationship is linear and/or shift-invariant.

Problem 2.4 For each of the following pairs of sequences, x(n) represents the input to an LSI system with unit-sample response h(n). Determine each output y(n). Sketch your results.

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Problem 2.5 The system shown below contains two linear shift-invariant subsystems with unit sample responses h1 (n) and h2 (n), in cascade.

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(a) Let x(n) = u(n). Find ya (n) by first convolving x(n) with h1 (n) and then convolving the result with h2 (n) i.e.

(b) Again let x(n) = u(n). Find yb(n) by convolving x(n) with the result of the convolution of h1 (n) and h2 (n) i.e.

Your results for parts (a) and (b) should be identical, illustrating the associative property of convolution Problem 2.6 If the output of a system is the input multiplied by a complex constant then that input function is called an eigen function of the system. (a) Show that the function x(n) = zn, where z is a complex constant, is an eigen function of a linear shift-invariant discrete-time system. (b) By constructing a counterexample, show that z nu(n) is not an eigen function of a linear shift-invariant discrete-time system Matlab Assignment Experts

Solution 2.1 x(n) is periodic if x(n) sequence in (a),

=

x(n + N) for some integer value of N. For the

x(n + N) = x(n) if 7N is an integer multiple of 27. The smallest value of N for which this is true is N = 14. Therefore the sequence in (a) is periodic with period 14.

For the sequence in (b),

The factor ej8-is unity for (N/8) an integer multiple of 2rr.

This requires that where N and R are both integers. This is not possible since 7 is anirrational number. Therefore this sequence is not periodic.

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Solution2.2

Solution 2.3 Each of the systems given can be tested against the definitions oflinearity and time invariance. For example, for

The system is not linear. The system is, however, shift-invariant since T[x(n-n0 )J = 2x(n-n ) + 3 = y(n-n ) In a similar manner we can show that: (b) is linear but not shift-invariant (c) is not linear but is shift-invariant (d) is linear and shift-invariant Matlab Assignment Experts

Part (b) can likewise be done graphically. Alternatively since

Since 6(k + 2) = 0 except for k = -2, and is unity for k = -2 y(n) = x(n + 2).

For part (c) x(k) and h(n -k) are as sketched below: Matlab Assignment Experts

Graphically we see that for n < 0, x(k) h(n -k) is zero and consequently y(n) = 0, n < 0. For n > 0

Consequently for all n, which is a decaying exponential for n > 0.

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The answer for part (d) is:

Solution 2.5 Therefore with x(n) as a unit step, x(n) * h (n) is:

Convolving w(n) graphically with h2 (n)

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The convolution of h1 (n) and h2 (n) is: h(n) = h1 (n) * h 2 (n) = (.8)n u(n) -(.8) (n-3) u(n -3)

The convolution of this result with a unit step results in

Or,

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Solution 2.6 The fact that x(n) = zn is an eigen function follows from the convolution sum. Specifically

Since the summation in the equation (S2.6-1) does not depend onn, it is simply a constant for any given z.While the complex exponential z n is an eigenfunction of any linear shift-invariant system, zn u(n) is not. For example, let h(n) = 6(n1). Then with x(n) = zun u(n), y(n) = zn-1 u(n-1), which is not a complex constant times x(n).

S2.6 Matlab Assignment Experts