WHAT DOES FILTERED WHITE NOISE LOOK LIKE?

This is just an intuition-building demo. There's no new theory.

We'll pass white noise x(t) through a lowpass or bandpass filter. What does a typical sample function of the output y(t) look like? How is its power affected? How quickly does it vary in time?

To find out, we'll just do it and look at the result. Method: (1) generate vector x(t) of white noise samples; (2) FFT it to produce X(f) (actually, this is unnecessary, since transformed white noise is white in frequency, too); (3) multiply by the filter H(f) to obtain Y(f); (4) inverse FFT to obtain y(t). These steps take up space on the page, so they are contained in a hidden area farther below.

The section below is interactive if you are reading this with Mathcad. If you are not, then review some of the precalculated examples that follow.

First, select the low and high edges of the filter passband:

Frequency is normalized by the sampling rate, so it runs from -0.5 to 0.5.

Example: An All-Pass Filter

Since the filter is "wide open," the output in this case equals the input x(t) - that is, the white noise.

Example: Lowpass Filter

Now that the bandwidth is reduced, the filter output varies more slowly and it has less power (lower variance). Its power is proportional to the total filter bandwidth.

Example: Lowpass Filter With a Small Bandwidth

Here the power is down to about 10% of that of the input. Variations are even slower.

Example: Bandpass Filter

Here we have removed DC and very low frequencies, so the output appears more oscillatory - it can't stay away from zero volts, since DC is gone. The power is less than that of the input, because the filter has lower bandwidth.

Example: Bandpass Filter, Narrowband

This one is interesting. Its spectrum is so narrow about a frequency of 0.25 (four samples per period) that it behaves like a sine wave of that frequency, but with randomly varying amplitude and phase. The variability in envelope (amplitude, phase) is determined by the bandwidth of the filter.

Example: Bandpass Filter, Very Narrowband

Very slow variation! That's because the bandwidth is so small. Also the power is greatly reduced, for the same reason.

Example: Bandpass With Different Centre Frequency

This one has the same bandwidth, hence the same rate of variation, as the last example. The difference is the lower centre frequency, so the oscillations are slower.