Sine wave in frequency domain on signal analyzer

Question

The Fourier Transform of sine wave ($\sin(2 \pi A t)$) is given as : $$\frac{1}{2i} [\delta(f-A)-\delta(f+A)]$$

This means that the Fourier Transform of the real function, $\sin(t)$ has an imaginary Fourier Transform (no real part).

How can we observe this Fourier Transform of sine wave on signal analyzer if it is imaginary with an "$i$" in it?

The thing is that I have a very high frequency signal (a few GHz) and so I am not able to see this signal clearly in time domain because the resolution of my signal analyzer is not that good. So I was trying to view it in frequency domain instead of time domain. Right now, in frequency domain I am able to view this signal as a single spike. So, should I conclude that the signal generated is a sine wave?

Answer 1

If the instrument you are using to measure is not a vector analyzer (i.e. it does not measure both phase and magnitude), then the most probable thing is that you are measuring the magnitude of the Fourier transform only. So the fact that, mathematically, the transform is purely imaginary does not matter that much in terms of measuring.

Regarding the single spike, remember that any real signal in the time domain has a magnitude such that

$$x(t)\in \mathbb{R} \implies|X(j\omega)|=|X(-j\omega)|$$

That means that if the signal is real (as in this case), then the magnitude of the Fourier transform is even. In other words, if we know that the signal is real in the time domain and we just want to know the magnitude of the Fourier transform, then the transform for $\omega <0$ doesn't give us any new information: all we need is in the positive $\omega$-axis.

So yes, if you are seeing a single spike in your analyzer, then you are indeed looking at the transform of a sinusoidal wave.

Answer 2

in frequency domain each spike/impulse represents to a sinusoid in time domain. so if in frequency domain the signal shows a single spike/impulse it means the input signal has a sinusoid in time domain.

Answer 3

The fourier transform has both phase and magnitude information. The complex scaling factor you mention only gives information about the phase, whereas the output on the signal analyzer is presumably giving you the magnitude estimate. It comes back to this picture, and might help to consider the case of cosine instead of sine:

Answer 4

In terms of a real signal being input to a signal analyzer or FFT, an imaginary signal is just a real signal with a 90 degree phase shift relative to some reference.

A complex signal (with a potentially non-zero imaginary component) is just a way to represent a phase shift relative to a reference cosine sinusoid.