Question

If \(f(t) = [-1, 2, 1]\) and \(g(t) = [0, -1, 2]\) are two wavelets and \(f(\tau)\) and \(g(\tau)\) represent flipped versions of \(f(t)\) and \(g(t)\), respectively, then which of the following expressions is/are CORRECT?
[* and \(\otimes\) denote convolution and cross-correlation operations, respectively]

• A. \(f(t)*g(t)=f(t)\otimes g(t)\)
• B. \(f(t)*g(t)=f(t)\otimes g(\tau)\)
• C. \(f(t)*g(t)=f(\tau)\otimes g(t)\)
• D. \(f(t)*g(t)=f(\tau)\otimes g(\tau)\)

Hint:

Convolution equals cross-correlation once EXACTLY ONE of the two signals is time-reversed (flipped); flipping both or neither does not reproduce the convolution.

Answer 1

The Correct Option is B, C

Convolution reverses (flips) one of the two sequences before sliding and multiplying, whereas cross-correlation slides the two sequences past each other WITHOUT reversing either one. Comparing the two definitions directly gives the general identity: convolving \(f\) and \(g\) is the same as cross-correlating either signal, PROVIDED the OTHER signal is flipped first - flipping just one of the two signals before cross-correlating reproduces the convolution.

Let us check this with the numbers given. First compute the full linear convolution:
\(f(t)=[-1,2,1]\), \(g(t)=[0,-1,2]\)
\[f*g=[0,\;1,\;-4,\;3,\;2]\]
(term by term: \((-1)(0)=0\); \((-1)(-1)+(2)(0)=1\); \((-1)(2)+(2)(-1)+(1)(0)=-4\); \((2)(2)+(1)(-1)=3\); \((1)(2)=2\)).

Now flip \(f\): \(f(\tau)=[1,2,-1]\). Cross-correlating \(f(\tau)\) with the unflipped \(g(t)\) (sliding \(g\) past \(f(\tau)\)) reproduces exactly:
\[f(\tau)\otimes g(t)=[0,\;1,\;-4,\;3,\;2]\]
which matches \(f*g\) term for term - so option (C) is correct.

Similarly, flip \(g\) instead: \(g(\tau)=[2,-1,0]\). Cross-correlating the unflipped \(f(t)\) with \(g(\tau)\) also reproduces the same convolution sequence \([0,1,-4,3,2]\) - so option (B) is correct as well.

By contrast, cross-correlating \(f\) and \(g\) with NEITHER of them flipped gives \([0,-1,0,5,-2]\), which is a completely different sequence from \(f*g\), so option (A) is wrong. And cross-correlating \(f(\tau)\) with \(g(\tau)\) (BOTH flipped) reproduces \([2,3,-4,1,0]\), which is just the time-reversed version of \(f*g\) rather than \(f*g\) itself, so option (D) is also wrong.

Hence the correct expressions are \(f(t)*g(t)=f(t)\otimes g(\tau)\) and \(f(t)*g(t)=f(\tau)\otimes g(t)\), i.e. options \(\boxed{(B)\ \text{and}\ (C)}\), answer 2,3.