EC-7.4 Frequency Response: What a System Does to Each Component

What this is and why it exists

The Fourier transform was defined in the transforms module. Its properties were derived there and you learned how to compute one. This topic is not that.

This topic is about what the transform of an impulse response tells you. That is a different question, and it has a different answer. It tells you what the system does to each frequency, separately, and it is the description an engineer actually reasons with.

Three later subjects lean on this for the same reason. An amplifier is specified by its bandwidth and its gain. A control loop is judged by its margins on a frequency plot. A communication channel is described by the band it passes. In each case the object being discussed is the one built here.

The vocabulary

• Eigenfunction — an input a system passes through unchanged in shape, altered only by a scale factor.
• Frequency response — the complex scale factor as a function of frequency. It is the Fourier transform of the impulse response.
• Magnitude response — the size of that factor, saying how much each frequency is amplified or attenuated.
• Phase response — the angle of that factor, saying how much each frequency is delayed.
• Passband — the range of frequencies a filter keeps.
• Stopband — the range it removes.
• Decibel — a logarithmic measure of a ratio, used for the magnitude axis.
• Group delay — the rate of change of the negative phase with frequency; the delay a whole waveform experiences.
• Distortionless — passing a signal with its shape unchanged, allowing only a scale and a delay.

The mental model

Begin with the observation everything rests on. Feed a complex exponential of some frequency into a linear time-invariant system. What comes out is the same complex exponential, multiplied by a complex number. Not a different frequency. Not a different shape. The same exponential, scaled.

This follows directly from the last topic. Convolving a complex exponential with the impulse response gives the exponential back, multiplied by an integral that no longer depends on time. That integral is the scale factor, and it depends on the frequency you chose.

Such an input is called an eigenfunction of the system. A whole family of them exists, and that is what makes frequency the natural coordinate here. In time, a system smears its input in a way that is awkward to picture. In frequency, it does something almost embarrassingly simple: it multiplies each component by a number.

Collect those numbers over all frequencies and you have the frequency response. It is exactly the Fourier transform of the impulse response, which the transforms module already showed you how to compute. Same system, second description, and the two carry identical information.

The response is complex, so it splits into a size and an angle, and both matter.

The magnitude says how much each frequency is amplified or attenuated. This is the half people remember, and it is what a specification sheet shows.

The phase says how much each frequency is delayed. This half gets ignored and should not be. A system with a perfectly flat magnitude can still ruin a waveform. Delaying the components by different amounts pulls the shape apart, while leaving the amount of energy at each frequency untouched. Anyone who has heard a poorly filtered audio channel has heard this.

Now the reason for the whole exercise. In the last topic, output came from input by convolution, which was awkward. Here, convolution becomes multiplication: the transform of the output is the transform of the input multiplied by the frequency response. A sliding overlap integral has been replaced by a product.

That trade is the engine of transform methods everywhere. You pay a transform on the way in and an inverse transform on the way out. In exchange, the operation in the middle becomes trivial. The trade is overwhelmingly worth it when the operation in the middle happens many times. It is also worth it when you care about the shape rather than one particular answer.

Filters are now nameable. A filter is a system chosen for the shape of its magnitude response, and the names say which frequencies survive. Low pass keeps the low ones and high pass the high. Band pass keeps a middle range, and band stop keeps everything except a middle range.

The ideal versions of these are worth meeting because of how they fail. Take a magnitude response that is exactly one in the passband and exactly zero outside it, with a vertical edge. Transform it back and the impulse response spreads out in both directions and never quite ends. It is nonzero before the origin, so the perfectly sharp filter is not causal and therefore cannot be built. This is the first honest limit the subject imposes. It is why real filters are specified with a transition width and a tolerance rather than an edge.

Two practical matters remain.

Responses are plotted with magnitude in decibels against a logarithmic frequency axis. The reason is not tradition. On those axes the response of most systems becomes a set of straight line segments meeting at corners. Each corner sits at a frequency where something in the system changes behaviour. Sketching such a plot from a system description is a skill the control and amplifier work will assume you have. So is reading a system description back off the plot.

Finally, the precise meaning of distortion. A system passes a signal through with its shape unchanged only under two conditions. Its magnitude must be constant across the band the signal occupies, and its phase must change linearly with frequency there. Constant magnitude scales everything equally. Linear phase delays everything equally, which moves the waveform without deforming it. Departures from those two conditions are what distortion is, and stating it this way turns a vague complaint into a measurement.

What you should now be able to explain or do

• Explain why a complex exponential comes out of such a system unchanged in shape, and what that has to do with the impulse response.
• Say what the frequency response is and how it relates to the impulse response.
• Describe separately what the magnitude and the phase do to a signal, and give a case where phase alone ruins a waveform.
• State why convolution becoming multiplication is the reason to change domain.
• Name the four filter shapes and explain why the perfectly sharp version cannot be built.
• Sketch and read a magnitude response on decibel and logarithmic axes.
• Give the two conditions for distortionless transmission and say what each one prevents.

Check yourself