Syllabus: LTI systems: definition and properties, causality, stability, impulse response, convolution, poles and zeroes, frequency response, group delay, phase delay.
12 hand-picked visual resources on this topic, 6 of them little-known gems — under each card, look for See it. Every link is checked against the live site before it ships.
Test each property separately and mechanically. For time invariance, compare the output of a delayed input against the delayed output — if the time axis is warped rather than shifted, the system is time varying.
Stability is a statement about where the region of convergence lies, not just about pole positions. For a causal system the ROC is outside the outermost pole; stability then needs every pole inside the unit circle.
A sinusoid into an LTI system comes out as the same sinusoid scaled by ∣H(jω)∣ and shifted by ∠H(jω) — nothing else. Distortionless transmission therefore needs flat magnitude and phase linear in ω; a constant phase is not good enough.
x(t)=cosω0t⇒y(t)=∣H(jω0)∣cos(ω0t+∠H(jω0))
steady state only
τp=−ω∠H(jω),τg=−dωd∠H(jω)
distortionless: ∣H∣=K,∠H=−ωt0⇒y(t)=Kx(t−t0)
linear phase⇔constant group delay⇔no dispersion
why FIR filters with symmetric coefficients are used for data
H(jω)=F{h(t)},∣H(−jω)∣=∣H(jω)∣
Where marks are lost
Group delay is the delay of the envelope, phase delay that of the carrier. They are equal only when the phase is linear through the origin — a constant phase offset makes them differ.
The impulse response h(t)is the system — every property in this topic is a statement about it. The step response is its running integral, which is why a question giving one and asking for the other is a single differentiation or integration.
s(t)=∫−∞th(τ)dτ,h(t)=dtds(t)
discrete: s[n]=∑k≤nh[k], h[n]=s[n]−s[n−1]
causal⇔h(t)=0fort<0
BIBO stable⇔∫∣h(t)∣dt<∞
absolutely integrable — not merely decaying
memoryless⇔h(t)=Kδ(t)
cascade: h1∗h2;parallel: h1+h2
H(s)=L{h(t)},H(jω)=F{h(t)}
the transfer function is the transform of the impulse response
Where marks are lost
1/t decays to zero and is still not absolutely integrable, so decaying is not the same as stable. Check the integral, not the tail.
The transform turns differentiation into multiplication by s, which is the whole reason circuits and systems are solved in this domain. Learn the six standard pairs and the shift rules; almost every question is one of those plus partial fractions.
Transfer functions compose by simple algebra, which is why the s domain is worth the trip. Cascading multiplies, paralleling adds, and feedback divides — and the ROC of a cascade is at least the intersection of the parts.
cascade: H1H2,parallel: H1+H2
impulse responses convolve and add respectively
feedback: 1+H1H2H1
inverse system: Hinv(s)=H(s)1
exists and is stable only if H is minimum phase
ROCcascade⊇ROC1∩ROC2
a pole–zero cancellation can enlarge it
Energy and power signals
A signal is one or the other, never both: finite non-zero energy means zero average power, and finite non-zero power means infinite energy. Periodic signals are always power signals; a decaying exponential is an energy signal.
Convolution and Unit Impulse Responsehidden gemPhysics Videos by Eugene Khutoryansky · 9 minThe output built from scaled, shifted impulse responses, stacked up in 3D.