Impulse Decomposition

Representation of the Discrete-Time Signal x[n] using Impulses

It produces a shifted impulse response h[n - k]

What happens when an LTI system is applied to δ[n - k]

Convolution Sum Formula

When the input has short duration (few nonzero samples).

When is direct convolution by summing shifted impulse responses most effective?

Write the input as a weighted sum of shifted impulses.

First step in direct convolution?

x[k]h[n - k]

What is the output corresponding to x[k]δ[n - k]

By summing all weighted, shifted impulse responses.

How is the final output obtained in direct convolution?

y[n] = x[n] + (1/2)x[n - 1]

The system equation for the two-path multipath channel

The impulse response of the system equation for the two-path multipath channel

Example 2.1

It requires summing many shifted impulse responses for each n

Why is direct convolution inefficient for long signals?

w_n[k] = x[k]h[n - k]

What is the intermediate signal used in convolution evaluation?

Because n is treated as a constant during summation over k

Why is n written as a subscript in w_n[k]?

Expression of the output using the Intermediate Signal

Because h[n - k] = h[-(k - n)], which involves reflection in k

Why is the impulse response reflected in convolution?

Reflect about k = 0, then shift by n

What are the two operations applied to h[k] inreflect-and-shift convolution?

Shifted to the left

If n < 0, how is h[n - k] positioned?

Shifted to the right

If n > 0, how is h[n - k] positioned?

When the nonzero portions of x[k] and h[n - k] do not overlap

When is w_n[k] = 0 for all k?

Because the mathematical form of w_n[k] changes only when overlap changes.

Why do we analyze convolution over intervals of n?

The start or end of overlap between x[k] and h[n - k]

What determines the boundaries between intervals of n?