ELEC2134 Chap.8 Convolution and Circuit Response
Convolution and Circuit Response
An LTI circuit can be described by its impulse response in time or its transfer function in frequency. The output is the convolution of input and impulse response, while its transform is the product of the input spectrum and transfer function.
This chapter uses support overlap to derive piecewise responses, then applies magnitude and phase at one frequency to find sinusoidal steady state.
Convolution becomes manageable when treated as geometry before algebra. Write where x(tau) and h(t minus tau) are nonzero, solve the overlap inequalities and mark the times at which the limits change.
The resulting response is piecewise because the overlap can grow, remain complete, then shrink. Boundary continuity and support provide checks independent of integration.
The frequency-domain route gives a complementary interpretation. Transfer-function magnitude scales each sinusoidal component and transfer-function phase shifts it.
A single sinusoid keeps its frequency through an LTI circuit, while a cascade multiplies the component transfer functions. DC gain, causality and stability connect these formulas to physical behaviour and reveal results that are mathematically neat but electrically implausible.
What this chapter covers
- 01
Linearity and time invariance
- 02
Impulse response as a complete system description
- 03
Convolution integral and support overlap
- 04
Transfer function and spectral multiplication
- 05
Sinusoidal steady-state gain and phase
- 06
Cascade systems and multiplication of transfer functions
Worked example · free
Convolve a finite pulse with a causal response
- +2The pulse requires 0 less than or equal to tau less than or equal to 1, while h(t - tau) requires tau less than or equal to t.
- +2For t below zero there is no overlap. For 0 less than or equal to t below 1, integrate from zero to t to obtain y(t) = 1 - exp(-2t).
- +3For t at least 1, integrate from zero to 1 to obtain y(t) = exp[-2(t - 1)] - exp(-2t).
- +1Both active formulas equal 1 - exp(-2) at t = 1, confirming continuity.
Key terms
- LTI system
- A system satisfying linearity and time invariance, enabling impulse-response analysis.
- Impulse response
- The output h(t) caused by a unit impulse input.
- Convolution integral
- The sliding overlap integral combining an input with an impulse response.
- Transfer function
- The frequency response H(omega) that multiplies the input spectrum to produce output spectrum.
- DC gain
- The transfer-function value at zero frequency, describing response to constant input.
Convolution and Circuit Response FAQ
Why does convolution often need a piecewise answer?
As one function slides across another, their nonzero supports begin to overlap, may reach a full-overlap interval, and then separate. The valid integration limits change at those boundary times. Each overlap regime therefore produces a different expression, even though all pieces come from the same convolution integral.
When is the frequency-domain route easier?
Use Y(omega) = X(omega)H(omega) when transform pairs are known, when the circuit transfer function is already given, or when the input is sinusoidal. Time-domain convolution is often clearer for short pulses and simple support geometry. Both routes are equivalent and can be used to check one another.
What happens to a sinusoid through an LTI circuit?
A sinusoid is an eigenfunction of an LTI system. The steady-state output keeps the same angular frequency, while its amplitude is multiplied by |H(omega0)| and its phase is shifted by angle H(omega0). A result containing a new frequency signals nonlinearity or an algebra mistake.
How do I find convolution breakpoints?
Write the support inequalities for x(tau) and h(t minus tau), then solve where their interval endpoints first meet, fully overlap or separate. Those event times become the piecewise boundaries. Sketching the intervals before integrating prevents one pair of limits from being used across incompatible overlap regimes.
What does unity DC gain mean?
A transfer function with H(0) = 1 passes the steady value of a constant input without amplitude change after transients settle, assuming stability. It does not mean every frequency has gain one. The circuit may still attenuate higher-frequency components and add frequency-dependent phase.
Exam move
For every convolution problem, mark where x(tau) and h(t - tau) are nonzero, solve the inequalities and sketch overlap length before integrating. For a sinusoidal input, evaluate H only at the input frequency and report both gain and phase. Check causality, boundary continuity, output area and DC gain. Practise switching domains so you can choose the shorter route under exam time pressure.
Use one colour for input support and another for the shifted response, then state each interval before calculating. For cascades, multiply transfer functions first and inspect limiting gain before expanding any algebra. Revisit one pulse-response problem after changing a support boundary, because that variation tests whether the limits came from reasoning or memory.
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