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Suppose that I have two signals $x[n] = \left\{2,4,1\right\}$ and $p[n] = \left\{5,1,8\right\}$ and I want to multiply them.

  • How do you do that?
  • How different is it from convolving two signals?

I understand that multiplication in one domain is equal to convolution in other domain. How do you choose as to what to use : multiplication or convolution?

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    $\begingroup$ How different are apples and oranges? This part of the question is too broad, please specify what kind of difference interests you. $\endgroup$
    – chirlu
    Aug 23, 2013 at 7:02
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    $\begingroup$ This is like asking "What's better, an umbrella or a hammer?" Depends what you're trying to do. $\endgroup$
    – endolith
    Aug 23, 2013 at 14:19
  • $\begingroup$ Will Normal Time domain convolution result in frequency domain multiplication ? or it should be CIRCULAR CONVOLUTION ! Somewhere I read Property of Discrete Fourier Transform [DFT] if x(n) and h(n) are two time-domain signals, circular convolution between them x(n)©h(n) is equivalent to multiplication in the frequency domain. Because this property used in cyclic prefix for OFDMA systems. $\endgroup$
    – rajez79
    Aug 26, 2013 at 16:45

1 Answer 1

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Yes, you are correct. Multiplication in time domain means convolution in frequency domain and vice versa. Multiplying your signals $x[n]$ and $y[n]$ will give an output:

\begin{align} z[n]&=\{2\cdot 5, 4\cdot 1, 1\cdot 8\}\\ &= \{10, 4, 8\}\end{align}

Remember that this output is in time domain. When you convolve $x[n]$ and $y[n]$, you will get $z[n]$ in time domain as:

\begin{align} z[n]&=\{5\cdot 2, 5\cdot 4+1\cdot 2, 5\cdot 1+1\cdot 4+8\cdot 2, 1\cdot 1+8\cdot 4, 8\cdot 1\}\\ &= \{10,22,25,33,8\} \end{align}

Just flip one of the signals around zero and start moving right one place at a time. Multiply the corresponding points as you go along. The output has a larger sequence because convolution output has ${\rm length}(x)+{\rm length}(y)-1$ points.

To answer your query about where to use multiplication & convolution, assume you want to pass signal $x(n)$ through filter $y(n)$. The output of the filter $z(n)$ will be convolution of $x(n)$ and $y(n)$.
Now assume that you first converted from time to frequency domain, i.e. $X(e^{i\omega})$ and $Y(e^{i\omega})$ are frequency domain representation of $x(n)$ and $y(n)$. Now, to find $Z(e^{i\omega})$(output in frequency domain), you have to multiply $X(e^{i\omega})$ and $Y(e^{i\omega})$. To get the output in time domain i.e. $z(n)$ you have to apply inverse transform.

So that's how you use convolution and multiplication.

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    $\begingroup$ So when to use multiplication and when to use convolution? From what you explained convolution is equal to polynomial multiplication. $\endgroup$ Aug 23, 2013 at 7:24
  • $\begingroup$ @user2381442 edited the answer to include your query. And yes, convolution is a polynomial multiplication operation $\endgroup$ Aug 23, 2013 at 8:17
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    $\begingroup$ Convolution is filtering in the spatial domain, multiplication is filtering in the Fourier domain. What to use depends on what you are trying to do. $\endgroup$ Aug 23, 2013 at 8:17

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