ELEC 243 Lab

Interlude

The Spectrogram

The Fourier series describes a periodic signal by representing it as a weighted sum of harmonics of the fundamental frequency ($f_0$ ):

$\displaystyle x(t)=\sum_n c_n e^{jn2\pi f_0t}$ .
The Fourier coefficients ($c_n$ ) specify how much of the corresponding frequency ($nf_0$ ) is present in the signal.

The Fourier transform extends this concept to arbitrary signals by considering frequency to be a continuous variable:

$\displaystyle x(t)=\int^\infty_{-\infty} X(f) e^{j2\pi ft} df$
For periodic signals, $X(f)=0$ everywhere except for $f=nf_0$ and we get the Fourier series (after some mathematical juggling).

$X(f)$ (or $c_n$ for periodic signals) is called the spectrum of $x(t)$ and describes the signal in terms of its frequency content. Since the Fourier transform is invertable, we can use either $x(t)$ or $X(f)$ to represent the signal.

This is an extremely useful tool if we're going to try to separate signal from noise by filtering. If we know the spectra of the signal and the noise we can try to design a filter which passes as much of the former and rejects as much of the latter as possible.

Since both the transfer function and the spectrum are plotted with the same x-axis (frequency) we can simply overlay the frequency response of a filter on the spectrum of a signal to see the effect. This makes it easy to visually evaluate prospective filters.

Here's an example of a spectrogram, in this case of a 2 kHz sine wave.

We would expect to see a single line at 2 kHz and zero everywhere else. In fact, due to the inevitable noise there's a very small but non-zero signal everywhere else. As an aside, since the y-axis is logarithmic (dBV means "decibels with respect to 1 volt"), zero would be at $-\infty$ .