Convolution with time shifted step function.

[seang]

How do you go about convolving when x(t) = u(t-1)? Can you just make the limits of integration 1 to t instead of 0 to t?

 

[abdo375]

Yes this is what it does, the only use of the unit-step function inside the integral is to change the limits

 

[ravenprp]

When x(t) is a time shifted step function, u(t-1), the convolution process can be simplified by using the time shifting property of the convolution integral. This property states that convolving a signal with a time shifted version of itself is equivalent to shifting the resulting convolution integral by the same amount.

In this case, convolving x(t) with u(t-1) can be achieved by shifting the limits of integration from 0 to t to 1 to t. This is because the step function u(t-1) is equal to 0 for t < 1 and 1 for t ≥ 1. Therefore, integrating from 0 to t will result in 0 for the first interval and the integral from 1 to t for the rest of the intervals, which is equivalent to convolving with u(t-1).

So yes, you can make the limits of integration 1 to t instead of 0 to t when convolving with a time shifted step function. This simplifies the convolution process and makes it easier to calculate.