Question

Problem 3. (LTI systems, impulse response) Recall that any signal can be decomposed into a sum of scaled unit impulses and recall that a unit impulse can be constructed from a linear combination of unit step functions. Now consider ?[n] = a^n u[n], 0 < a < 1 a. Show that any signal x[n] can be decomposed as x[n] = ?_{k=-?}^? c_k ?(n - k) and express c_k in terms of x[n]. Hint: recall that ?[n] = u[n] - u[n - 1]. Show that ?[n] = ?[n] - a?[n - 1] and use this to answer part a. b. Use the properties of linearity and time invariance to express the output y[n] = A(x[n]) in terms of the input x[n] and the signal g[n] = A(?[n]), where A(·) is an LTI system. c. Express the impulse response h[n] = A(?[n]) in terms of g[n] = A(?[n]).

          Problem 3. (LTI systems, impulse response) Recall that any signal can be decomposed into a sum of scaled unit impulses and recall that a unit impulse can be constructed from a linear combination of unit step functions. Now consider

?[n] = a^n u[n], 0 < a < 1

a. Show that any signal x[n] can be decomposed as

x[n] = ?_{k=-?}^? c_k ?(n - k)

and express c_k in terms of x[n].
Hint: recall that ?[n] = u[n] - u[n - 1]. Show that ?[n] = ?[n] - a?[n - 1] and use this to answer part a.

b. Use the properties of linearity and time invariance to express the output y[n] = A(x[n]) in terms of the input x[n] and the signal g[n] = A(?[n]), where A(·) is an LTI system.

c. Express the impulse response h[n] = A(?[n]) in terms of g[n] = A(?[n]).

Problem 3. (LTI systems, impulse response) Recall that any signal can be decomposed into a sum of scaled unit impulses and recall that a unit impulse can be constructed from a linear combination of unit step functions. Now consider

?[n] = a^n u[n], 0 < a < 1

a. Show that any signal x[n] can be decomposed as

x[n] = ?k=-?^? ck ?(n - k)

and express ck in terms of x[n].
Hint: recall that ?[n] = u[n] - u[n - 1]. Show that ?[n] = ?[n] - a?[n - 1] and use this to answer part a.

b. Use the properties of linearity and time invariance to express the output y[n] = A(x[n]) in terms of the input x[n] and the signal g[n] = A(?[n]), where A(·) is an LTI system.

c. Express the impulse response h[n] = A(?[n]) in terms of g[n] = A(?[n]).

Added by David M.

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