Question

The convolution sum is given by _____ equation.

I got this question by my school teacher while I was bunking the class.

I'm obligated to ask this question of Convolution : Impulse Response Representation for LTI Systems topic in section Time Domain Representation for LTI Systems of Signals and Systems

Answer 1

The convolution sum for discrete-time linear time-invariant (LTI) systems is given by the equation:

\[

y[n] = \sum_{k=-\infty}^{\infty} x[k] \cdot h[n - k]

\]

where:

- \( y[n] \) is the output signal,

- \( x[k] \) is the input signal,

- \( h[n - k] \) is the impulse response of the system,

- \( n \) and \( k \) are integer indices representing time steps.

In simpler terms, the convolution sum expresses how the output \( y[n] \) of an LTI system depends on the input signal \( x \) and the system's impulse response \( h \). It’s a fundamental concept for analyzing the behavior of LTI systems in the time domain.