EC-7.4 Frequency Response: What a System Does to Each Component
You can describe a linear time-invariant system by what it does to each frequency, read a magnitude and phase response, and say which filter a given response is.
Before:EC-4. Transforms, Complex Variables and Numerical MethodsUnlocks:S4-1. Analog CircuitsS4-3. Control SystemsS4-5. Probability Theory & Stochastic ProcessesS5-3. Digital Signal ProcessingPE2-3. Biomedical Instrumentation and Signal ProcessingEC-9. Microcontrollers and Embedded Systems
This is where the transform stops being mathematics and becomes a description of behaviour. EC-4 defines the Fourier transform and shows how to compute one; this topic is about what the transform of an impulse response tells you, which is a different question with a different answer. It is also the topic three later subjects all lean on for the same reason: an amplifier, a control loop and a communication channel are each described first by their frequency response.
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Complex exponentials as the signals such a system does not reshape
Feed a complex exponential into a linear time-invariant system and the same exponential comes out, scaled by a complex number. That property is why frequency is the natural coordinate for these systems and why every method here uses it.
MIT RES.6-007: Signals and Systems · CourseThe frequency response, as the transform of the impulse response
The complex scale factor at each frequency, taken together over all frequencies, is the frequency response, and it is the Fourier transform of the impulse response. The same system now has two equivalent descriptions.
NPTEL: Principles of Signals and Systems · CourseMagnitude and phase, and what each one does to a signal
The magnitude says how much each frequency is amplified or attenuated, and the phase says how much each is delayed. Both matter, and a system with a flat magnitude can still ruin a waveform through its phase.
NPTEL: Signals and Systems (IIT Kanpur) · CourseConvolution becomes multiplication, and why that is the whole reason to change domain
The awkward sliding operation of the previous topic becomes an ordinary product once both signals are described by frequency. Every transform method in engineering is bought with this one trade.
Filters: low pass, high pass, band pass, and the ideal that cannot be built
Naming a filter is naming which frequencies its magnitude response keeps. The perfectly sharp version of any of them turns out to be non-causal, which is the first honest limit this subject imposes.
NPTEL: Signals and Systems (IISER Bhopal) · CourseReading a response on logarithmic axes
Plotting magnitude in decibels against a logarithmic frequency axis turns the response of most systems into straight-line segments meeting at corners. Reading and sketching those segments is a skill the control and amplifier work assumes.
Distortionless transmission, and what it demands of the response
A system passes a signal through unchanged in shape only if its magnitude is constant and its phase changes linearly with frequency over the band in question. Departures from that are what distortion means precisely.
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