SC · Fourier

The Frequency Response

The frequency response H(ω) says what a system does to each frequency: the magnitude |H(ω)| is the gain — how much it amplifies or attenuates — and the phase ∠H(ω) is how far it shifts and delays. Read both directly from the sinusoid that comes out when you feed one in.

Which frequencies pass through

The top shows the input (faint) and output (bold) sinusoids at the same frequency. The output is the input scaled by |H| and shifted by ∠H. Below is the magnitude response |H(ω)| with a dot at the current frequency. Raise the frequency: low ones pass, high ones shrink and lag behind.

input sinusoidoutput: ×|H|, shift ∠H

magnitude |H(ω)|cutoff ωc · −3dB

input frequency ω4.00

Gain and phase

|H| = 0.45   ∠H = -63°

The output sinusoid has nearly vanished. This high frequency lies in the stopband the system blocks. The gain |H| is near 0 and the phase lags far behind.

Blocked

Probe the system with one sinusoid

In C4, a sinusoid was the eigenfunction of an LTI system. Feed in a sinusoid of frequency ω and out comes one of the same frequency, its size multiplied by |H(ω)| and its phase shifted by ∠H(ω). So measuring the amplitude ratio of the output sinusoid gives |H(ω)|, and its phase difference gives ∠H(ω), directly at that frequency. Gather this at every frequency and you have the whole frequency response H(ω).

Observecos(ωt) → |H| cos(ωt + ∠H)

Sinusoid in → same frequency, ×|H|, shift ∠H.

Chooselowpass |H(0)| = ?

A lowpass has gain 1 at DC.

Fill inat ωc: |H| = ?

At the cutoff the magnitude is 1/√2.

On your ownhigh ω: ∠H → ?

A first-order lowpass approaches −90° phase at high frequency.

Magnitude is gain, phase is delay

The magnitude |H(ω)| is the gain: above 1 it boosts that frequency, below 1 it cuts it. The phase ∠H(ω) is how far that frequency is shifted; a negative phase means the output comes out late — a delay. This lowpass system passes low frequencies almost untouched (gain 1, phase 0) and, the higher the frequency, attenuates it more (toward gain 0) and delays it more (toward phase −90 degrees).

Passband and stopband

The shape of the magnitude response is the filter’s type. The frequency band where |H| is near 1 is the passband; where it is near 0, the stopband. Their boundary is usually taken at the cutoff frequency ωc, where the magnitude falls to 1/√2 — that is, −3 dB. At this point the output power is half the input’s, so it is also called the half-power point. Lowpass, highpass, and bandpass differ only in where this passband sits. The next unit is the logarithmic plot that reads this |H| across a wide frequency range.

Back to the first screen

As you raised the input frequency, the output sinusoid shrank and lagged. How much it shrank was the magnitude |H(ω)| — the gain; how much it lagged was the phase ∠H(ω) — the delay. Low frequencies passed nearly intact (passband), high frequencies nearly vanished (stopband), and between them the −3 dB point where the magnitude falls to 1/√2 was the cutoff. The frequency response is the pair of plots — magnitude and phase — that records which tones a system passes and which it blocks.

The frequency response H(ω) = FT{h} is what an LTI system does to each frequency. The magnitude |H(ω)| is the gain (amplify or attenuate); the phase ∠H(ω) is the phase shift (delay). A sinusoid input of frequency ω becomes a sinusoid output at the same frequency, scaled by |H(ω)| and shifted by ∠H(ω) (eigenfunction). Where |H| is near 1 is the passband, near 0 the stopband, and where it falls to 1/√2 (−3 dB) is the cutoff ωc (half-power point).

On to the next unit

Drawing the magnitude on linear scales makes it hard to fit both the small differences in the passband and the steep rolloff of the stopband in one picture. The next unit, introduction to Bode plots, draws magnitude in decibels and frequency on a log axis. Then a wide frequency range fits at a glance, and a first-order system’s rolloff straightens into a line of −20 dB per tenfold frequency. A curved response simplifying into a few straight segments is the power of the Bode plot.