SC · Fourier

Bandwidth and Filtering

Filtering multiplies a signal’s spectrum by the filter’s frequency response. Frequencies in the passband survive; the stopband is removed. The width of the passband is the bandwidth, and the four filter types differ by where that passband sits. Match the passband to a target band yourself.

Match the passband to the target band

The faint flat line is an input spectrum holding all frequencies. The blue shape is the output after multiplying by the filter — only the passband survives. The gold dashed band is the target. Use the filter-type buttons and the cutoff slider to land the passband exactly on that band.

input (all frequencies)output = passbandtarget band

filter type

cutoff ωc2.00

Filter and passband

lowpass   ωc = 2.00

The passband differs from the target band. Only frequencies inside the passband survive; the rest are cut away. You must set both the filter type and the cutoff.

Off

Filtering is multiplying

The convolution theorem of C4 is the foundation of filtering. Passing a signal through a system is convolution in time, but in frequency it is multiplying the signal spectrum X(ω) by the filter response H(ω) point by point. Components in the passband, where H is near 1, survive intact; components in the stopband, where H is near 0, vanish. So the shape of a filter’s magnitude plot tells you at once which frequencies stay and which are cut.

ObserveY(ω) = X(ω) H(ω)

The output spectrum is input times filter.

Choosesurvives where |H| ?

The passband survives, the stopband vanishes.

Fill inlowpass bandwidth = ?

A lowpass bandwidth is the cutoff ωc.

On your ownfilter type = ?

The four filters differ by passband position.

Bandwidth is the passband width

Bandwidth is how wide the passband is, usually measured between the two points where the magnitude falls to −3 dB. For a lowpass it is from 0 to the cutoff ωc. A narrower bandwidth passes fewer frequencies and is more selective, but by the reciprocity of C3 the time response becomes slower and settling longer. Handling fast signals needs wide bandwidth; picking noise out narrowly needs narrow bandwidth. The two are a trade-off.

The four filters

Filters come in four kinds by where the passband sits. A lowpass passes low frequencies, a highpass high frequencies, a bandpass only a middle band, and a bandstop blocks only a middle band. To match the target band just now, the answer was the bandpass that passes only the middle. Real filters have a sloped transition band between passband and stopband, and the higher the order the steeper that transition, falling in multiples of −20 dB/decade. Only the shape differs; all work by the same principle — multiply the spectrum to keep just the frequencies you want.

Back to the first screen

To match the target band you had to pick the bandpass that passes only the middle and place that band with the cutoff. Multiplying the flat input spectrum by the filter left only the frequencies inside the passband; the rest were cut. The width of the surviving band was the bandwidth. Filtering is, in the end, one operation — multiply the spectrum by the filter’s shape — and lowpass, highpass, bandpass, bandstop are just where that passband sits. The convolution-is-multiplication property of C4 became a practical tool here.

Filtering multiplies a signal’s spectrum by the filter’s frequency response: Y(ω) = X(ω) H(ω), so passband frequencies (|H|≈1) survive and stopband ones (|H|≈0) vanish. The bandwidth is the width of the passband (usually between the −3 dB points); narrower is more selective but slower in time (C3 reciprocity). The four filters — lowpass, highpass, bandpass, bandstop — differ only by where the passband sits. Higher order steepens the transition to −20n dB/decade.

On to the next chapter

Fourier showed how a system handles steady-state sinusoids — its behavior along the ω axis. But the transient just after a system switches on, or an unstable divergence, is hard to capture with pure sinusoids alone. The next chapter, the Laplace transform and transfer function, widens the frequency jω into a complex s = σ + jω. Adding σ lets us handle growing and decaying exponentials too, so transients and stability show at a glance as the positions of poles and zeros on one plane. The a < 0 stability condition from B4 meets us again on that s-plane.