[問題] 一些信號與系統的問題

[joecheng (期末地獄)]

1.      (a) The impulse response h[n] of a linear time-invariant system is

            known to be zero, except in the interval N0≦n≦N1.

            The input x[n] is known to be zero,

            except on the interval N2≦n≦N3.

            As a result, the output is constrained to be zero,

            except in the some interval N4≦n≦N5.

            Determine N4 and N5 in terms of N0, N1, N2, and N3.

(b) If x[n] is zero, except for N consecutive points,

    and h[n] is zero, except for M consecutive points,

    what is the maximum number of consecutive points for which

    y[n] can be nonzero?

2.      By direct evaluation of the convolution sum,

        determine the step response of a linear time-invariant system

        whose impulse response is

          h[n]=a^(-n) u[-n]


3.      Consider an LTI system with frequency response

           H(e^jw)= (1-e^-j2w)/(1+1/2e^-j4w)  -pi<w<pi

Determine the output y[n] for all n if the input x[n] for all n is

           x[n]=sin(n pi/4)

4.      Consider a system with input x[n] and output y[n] that satisfy

        the difference equation

         y[n]=ny[n-1]+x[n]

The system is causal and satisfies initial-rest conditions;

i.e., if x[n] = 0 for n < n0, then y[n] = 0 for n < n0.

(a)     If x[n] = δ[n], determine y[n] for all n.
(b)     Is the system linear? Justify your answer.
(c)     Is the system time invariant? Justify your answer.


先謝謝大家

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