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SC · Fourier

The Frequency Response

The frequency response H(ω) records what a system does to each frequency. The magnitude |H(ω)| is the gain, how much each frequency is boosted or cut, and the phase ∠H(ω) is how much it is delayed; you read the two directly by feeding in one sinusoid and watching the sinusoid that comes out.

Which frequencies pass through

The upper figure shows input (faint) and output (gold) sinusoids of the same frequency, and the output is the input scaled by |H| and shifted by ∠H. The lower figure is the magnitude response |H(ω)| of a first-order lowpass system, with the dot at the current frequency. As the frequency rises, low frequencies pass almost unchanged, and past the cutoff frequency ωc = 2 the output shrinks and slips 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 has shrunk below half the input. This frequency lies toward the stopband the system blocks, and the higher the frequency, the closer the gain |H| gets to 0 and the closer the phase slips toward −90°.
Blocked

Probe the system with one sinusoid

In C4, sinusoids were the eigenfunctions of LTI systems. Feed in a sinusoid of frequency ω and a sinusoid of the same frequency comes out, its size multiplied by |H(ω)| and its phase shifted by ∠H(ω). So feeding in one sinusoid and measuring the amplitude ratio of the one that comes out gives |H(ω)| at that frequency directly, and measuring the phase difference gives ∠H(ω). Gathering these at every frequency gives the whole frequency response H(ω).

Observecos(ωt) → |H| cos(ωt + ∠H)
Sinusoid in → same frequency, ×|H|, shift ∠H.
Chooselowpass |H(0)| = ?
A lowpass system has gain 1 at DC.
Fill inat ωc: |H| = ?
At the cutoff the magnitude is 1/√2.
On your ownhigh ω: ∠H → ?
The high-frequency phase of a first-order lowpass is −90°.

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) shows how far that frequency is shifted, and a negative phase means a delay, the output coming out later. The first-order lowpass system on this screen has |H(ω)| = 1/√(1 + (ω/2)²) and ∠H(ω) = −arctan(ω/2), so it passes low frequencies with a gain close to 1 and a phase close to 0, while at higher frequencies the gain approaches 0 and the phase approaches −90°.

Passband and stopband

The shape of the magnitude response is the type of filter. The frequency range where |H| is close to 1 is the passband, the range where it is close to 0 is the stopband, and the boundary between them is usually taken as the cutoff frequency ωc, where the magnitude falls to 1/√2, that is, −3dB. At this point the output power is half the input, so it is also called the half-power point. Lowpass, highpass and bandpass filters all differ in where their passband lies, and the next unit is a logarithmic picture for reading this |H| over a wide frequency range.

Back to the first screen

As the input frequency rose, the output sinusoid grew smaller and slipped behind. How much it shrank was the magnitude |H(ω)|, the gain, and how far it slipped was the phase ∠H(ω), the delay. Low frequencies passed nearly unchanged (the passband), high frequencies were cut sharply (the stopband), and the −3dB point between them, where the magnitude falls to 1/√2, was the cutoff frequency ωc = 2. The frequency response is a picture, in two sheets, magnitude and phase, of which frequencies 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 (boost or cut), and the phase ∠H(ω) is the phase shift (delay). A sinusoidal input of frequency ω becomes an output sinusoid of the same frequency, with its size multiplied by |H(ω)| and its phase shifted by ∠H(ω) (eigenfunction). Where |H| is close to 1 is the passband, where it is close to 0 the stopband, and where it falls to 1/√2 (−3dB) is the cutoff frequency ωc (half-power point).
On to the next unit

Drawn on ordinary scales, the magnitude response struggles to show the small differences in the passband and the steep attenuation in the stopband in one picture. In the next unit, introduction to Bode plots, the magnitude is drawn in decibels and the frequency on a logarithmic axis. A wide frequency range then fits in a single view, and the attenuation of a first-order system straightens into a line of −20dB for every 10 times the frequency. Simplifying a curved response into a few straight pieces is the power of the Bode plot.