Convolve & Crosscorrelate Delayed Signals Example

[sagemode]

Homework Statement

Consider two signals gl(t) and g2(t). These two signals are delayed by amounts
equal to tl and t2 seconds respectively. Show that the time delays are additive
in convolving the pair of delayed signals, whereas they are subtractive in crosscorrelating
them. Explain with an example.

Homework Equations

I tried Convolution of delayed ex and u(t)

The Attempt at a Solution

Pretty much lost when I tried.

 

[KataKoniK]

Hello there,

I would like to provide some clarification on how time delays are handled in convolution and cross-correlation.

First, let's define convolution and cross-correlation. Convolution is a mathematical operation that combines two signals to produce a third signal. It is commonly used to find the output of a linear time-invariant system given its input and impulse response. On the other hand, cross-correlation is a measure of similarity between two signals as a function of the displacement of one relative to the other.

Now, let's consider the two signals gl(t) and g2(t), which are delayed by tl and t2 seconds respectively. When convolving these two signals, the resulting signal will be delayed by the sum of the two delays, i.e. tl + t2. This is because convolution is a linear operation, and the delay in one signal does not affect the delay in the other signal.

However, when cross-correlating the two signals, the resulting signal will be delayed by the difference of the two delays, i.e. tl - t2. This is because cross-correlation involves shifting one signal relative to the other, and the delay in one signal will affect the alignment of the two signals.

To better understand this concept, let's consider an example. Suppose we have two signals, x(t) and y(t), with time delays of tl = 2 seconds and t2 = 3 seconds, respectively. When we convolve these two signals, the resulting signal will have a time delay of tl + t2 = 5 seconds. However, when we cross-correlate these two signals, the resulting signal will have a time delay of tl - t2 = -1 second. This means that the two signals will be aligned in the opposite direction, with y(t) being shifted 1 second earlier than x(t).

I hope this helps to clarify the concept of time delays in convolution and cross-correlation. Let me know if you have any further questions or if you need any additional explanation.