The Heterodyning Interpretation

• When we studied ring modulation, we saw that the product of two sinusoids at frequencies $f_c$ and $f_m$ results in a signal with two spectral components at frequencies $f_c + f_m$ and $f_c - f_m$.

• If the modulating frequency $f_m = f_c$, the result will be two spectral components at frequencies of 0 and 2$f_c$.

• This result has interesting implications. If it was possible to design a very narrow low-pass filter that could essentially filter out all non-zero frequency components, we would have a means for estimating frequency content in a signal via modulation with a set of sinusoidal components, followed by the filtering operation.

• Taking another look at the formula for the DFT in Eq. (11), we see the complex exponential signals can be interpreted as modulating sinusoids.

• After modulation with the complex exponential signal (the point-by-point multiplication part of the inner product), the resulting signal values are summed (the last part of the inner product). This summation in effect computes the DC component of the signal. All non-zero frequency sinusoidal components sum to zero over an integer number of periods (as we saw at the beginning of this discussion).

• We can also look at this operation from a frequency-domain perspective. In this case, it should be pointed out that there is an implicit multiplication of $x[n]$ by a rectangular “window” (a sequence of ones of length $N$).

• Multiplication in the time domain corresponds to convolution (or filtering) in the frequency domain.

• Remembering the relationship between a unit impulse function and its corresponding spectrum, we see that the frequency-domain counterpart of a rectangular window is a “unit frequency impulse” centered at frequency bin $k=0$ (or DC).

• Thus, the DFT can be equally well understood to modulate each of the sinusoidal components of a signal $x[n]$ at frequencies $\hat{\omega}_k$ down to $\omega = 0$, where their relative strengths are then “extracted” by means of a very narrow lowpass filter.