SC · Fourier

Bandwidth and Filtering

Filtering means multiplying a signal's spectrum by a filter's frequency response. Frequencies inside the passband survive and those in the stopband vanish; the width of the passband is the bandwidth, and the four kinds of filter differ in where their passband lies. Choose a filter type and a cutoff and fit the passband onto the target band yourself.

Match the passband to the target band

The faint flat line is the input spectrum, with every frequency present equally, and the blue shape is the output after multiplying by the filter, so only the passband survives. The gold dashed band is the target, the stretch from 2.5 to 5. Use the filter-type buttons and the cutoff slider to fit the passband exactly onto that band; the filters in this figure are ideal, exactly 1 inside the passband and exactly 0 outside it.

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 live and the rest are cut away, so the filter type and the cutoff have to be set together.

Off

Filtering is multiplying

The convolution theorem of C4 is the foundation of filtering. Passing a signal through a system is convolution in the time domain, but in the frequency domain it is multiplying the signal spectrum X(ω) by the filter response H(ω) point by point. Frequency components in the passband, where H is close to 1, survive unchanged, and those in the stopband, where H is close to 0, vanish. So the shape of a filter's magnitude plot alone tells you at once which frequencies remain 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 means how wide the passband is, usually measured as the width between the two points where the magnitude falls to −3 dB. For a lowpass filter, the bandwidth runs from 0 up to the cutoff ωc. The narrower the bandwidth, the fewer frequencies are picked out and the higher the selectivity, but because of the reciprocal relation of C3 the response in the time domain becomes slower and takes longer to settle. Handling fast signals needs a wide bandwidth and filtering noise narrowly needs a narrow one, so the two are a trade-off.

The four filters

Filters come in four kinds according to where their passband lies. A lowpass passes low frequencies and a highpass high ones, a bandpass passes only one band in the middle, and a bandstop blocks only one band in the middle. The answer for fitting the target band just now was the bandpass, which passes only the middle. The filters in the figure are ideal shapes that jump between 1 and 0 at the edge in one step, but a real filter has a sloped transition band between passband and stopband, and the higher its order n, the steeper the transition, at −20n dB/dec. Only the shapes differ: all work on the same principle of multiplying the spectrum to keep only the wanted frequencies.

Back to the first screen

To fit the target band you had to choose the bandpass, which passes only the middle, and set the band's place with the cutoff. Multiplying the flat input spectrum by the filter left only the frequencies inside the passband and cut away the rest, and the width of the surviving band was the bandwidth. Filtering, in the end, is a single operation of multiplying the spectrum by the filter's shape, and lowpass, highpass, bandpass and bandstop differ only in where the passband is placed. The property from C4 that convolution is multiplication became a hands-on tool here.

Filtering is multiplying a signal spectrum by a filter's frequency response, Y(ω) = X(ω) H(ω). So frequencies in the passband (|H| ≈ 1) survive and those in the stopband (|H| ≈ 0) vanish. The bandwidth is the width of the passband (usually between the −3 dB points); narrower means more selective but a slower time response (the reciprocal relation of C3). The four filters, lowpass, highpass, bandpass and bandstop, differ only in where the passband lies, and the higher the order, the steeper the transition, at −20n dB/dec.

On to the next chapter

With Fourier's method we read how a system handles sinusoids in steady state, its behavior along the ω axis. But the transient response right after a system switches on, or an unstable blow-up, is hard to capture with pure sinusoids alone. The next unit, the Laplace transform, widens the frequency jω to the complex number s = σ + jω, and adding σ lets growing and decaying exponentials be handled too, so transient response and stability show at a glance as the positions of poles and zeros on one plane. The a < 0 stability condition seen in B4 also reappears on that s-plane.