Sentence examples for using a complex exponential from inspiring English sources

And what it says is, if you multiply by e to the plus j omega c t, then when you want to get back, multiply by e to the minus j omega c t. Now one question that you could conceivably be asking is, if we're talking about practical systems and not simply mathematics, does it make sense in the real world to consider using a complex exponential carrier?

The transform is calculated using a complex exponential basis and each component is normalized: begin{aligned} X[k] = frac{1}{N[k]} sum _{n=0}^{N[k]-1} x[n] e^{-frac{2 pi j Q n}{N[k]}}.

So what this says is that if we have a signal, x of t, and we use it to modulate a complex exponential carrier in the frequency domain, what we've simply done is to take the original spectrum and shift it in frequency.

Now more or less, the justification for this, or the proof, follows by simply looking at the response to a complex exponential, using the convolution integral.

That very often in practical systems one considers using a carrier which in fact is a complex exponential.

One of these into the system gives us, as an output, a complex exponential with the same complex frequency multiplied by what we refer to as the eigenvalue.

Well, in fact sinusoidal modulation, in other words, modulation using only a sinusoidal carrier, very often is used in its own right not only for generating a complex exponential carrier, but as a carrier by itself.

Well, a complex exponential is complex.

And what I mean by a complex exponential, again, is an exponential of the form C e ^ (a t).

And so, in fact, we put in a complex exponential, we get out a complex exponentials of the same frequency, multiplied by a complex constant.

And we modulate it with a complex exponential carrier with a carrier frequency, omega c.