Systems Properties & LTI Convolution

Time–Invariant (TI) Systems

Definition
- For every admissible input $x(t)\;(\text{or }x[n])$ and every time-shift $t0\;(\text{or }n0)$ ,
 $\text{System input }x(t-t0)\;\to\;\text{output }y(t-t0)$
 (analogously in DT).
Formal condition
- CT: $x(t-t0)\xrightarrow{\text{sys}}y(t-t0)\;\forall\,t_0\in\mathbb R$
- DT: $x[n-n0]\xrightarrow{\text{sys}}y[n-n0]\;\forall\,n_0\in\mathbb Z$
Test examples
- $y(t)=\sin(x(t))$ → TI ✔ (time enters neither explicitly nor via coefficients).
- $y[n]=n\,x[n]$ → TI ✘ (explicit dependence on $n$ violates invariance).
- $y[n]=x[n]x[n-3]$ → TI ✔ (all time indices shifted identically).
- $y(t)=x(5t)$ → TI ✘ (time scaling is not a pure shift).

Linear Systems

Definition (superposition)
- For any inputs $x1,x2$ and scalars $a1,a2\in\mathbb C$
 $x(t)=a1x1(t)+a2x2(t)\;\Rightarrow\;y(t)=a1y1(t)+a2y2(t).$
Test examples
- Integrator $y(t)=\int_{1}^{t}x(\tau)\,d\tau$ – Linear ✔ (limits independent of input; additive & homogeneous).
- $y[n]=(x[2n])^2$ – Linear ✘ (square breaks homogeneity).
- $y(t)=(t-1)x(t+2)$ – Linear ✔ (multiplying by external function of $t$ preserves linearity).

Discrete-Time LTI Systems

Impulse response & representation property

Define unit impulse $\delta[n]$ .
Impulse response: $h[n]=\text{output to }\delta[n].$
Any sequence can be decomposed:
$x[n]=\sum_{k=-\infty}^{\infty}x[k]\,\delta[n-k].$

Convolution sum

Via TI + linearity:
$y[n]=\sum_{k=-\infty}^{\infty}x[k]h[n-k]\;\triangleq\;(x*h)[n].$
Commutative form: $\sum{k}x[k]h[n-k]=\sum{k}x[n-k]h[k].$

Illustrative example (p. 5)

Given $h[n]=u[n]$ (sketched as 0.2,0.4,0.6,0.8,1) and input $x[n]=u[n]-u[n-3]$ (rectangular pulse of length 3).
Output obtained by convolution: perform shift-overlap-multiply-accumulate for each $n$ .

Continuous-Time LTI Systems

Pulse approximation argument

Rectangular pulse p_{\Delta}(t)=\begin{cases}1,&0\le t<\Delta\0,&\text{else}\end{cases}.
Approximate $x(t)\approx\hat x(t)=\sum{k=-\infty}^{\infty}x(k\Delta)\,p{\Delta}(t-k\Delta).$
Let $h{\Delta}(t)=\text{response to }p{\Delta}(t).$
By TI+linearity: $\hat y(t)=\sumk x(k\Delta)h{\Delta}(t-k\Delta).$
As $\Delta\to0$
- $p{\Delta}(t)\to\delta(t)$ , $h{\Delta}(t)\to h(t).$
- Riemann sum → integral ⇒
 $y(t)=\int_{-\infty}^{\infty}x(\tau)h(t-\tau)\,d\tau\;\triangleq\;(x*h)(t).$

Impulse response definition

$h(t)=\text{output to }\delta(t).$
Convolution integral as above.

Main Message

Knowing $h(t)\;(\text{or }h[n])$ fully characterises an LTI system:
$h \;\bigcirc\; x \;\longrightarrow\; y.$
Challenge: direct convolution can be algebraically heavy.

Identifying LTI Systems (p. 10)

a) $y[n]=3x[n]+2$ – Not LTI (fails linearity: constant term).
b) $y(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}x(\tau)\,[u(t-\tau)-u(t-\tau-1)]\,d\tau$ – LTI ✔ (integral kernel depends only on difference $t-\tau$ ).
c) $y(t)=x(2t)$ – TI ✘, so not LTI.
d) $y[n]=(x[2n])^2$ – Non-linear.

Computing Convolution (DT)

Algorithm (fix $n$ ):
1. Flip $h[k]\to h[-k]$ .
2. Shift by $n\;\to h[n-k]$.
3. Multiply with x[k] $sample-wise.</li><li>Sum over$ k $.</li></ol></li><li>Example (p. 11–15):<ul><li>$ h[n]=n(u[n]-u[n-3]),\;x[n]=u[n]-u[n-3]. $</li><li>Non-zero support 0–2. Convolution yields triangular shape of length 5, numerically:$ y[0]=0,\,y[1]=1,\,y[2]=3,\,y[3]=3,\,y[4]=1,\,\text{else }0. $</li></ul></li></ul><h3 id="4ddb6529-27b9-47ad-8fff-f34fd9bb5098" data-toc-id="4ddb6529-27b9-47ad-8fff-f34fd9bb5098" collapsed="false" seolevelmigrated="true">Computing Convolution (CT)</h3><ul><li>Formula$ y(t)=\int_{-\infty}^{\infty}x(\tau)h(t-\tau)\,d\tau. $</li><li>Procedure mirrors DT: flip$ h $, shift to$ t $, multiply with$ x $, integrate.</li><li>Example (p. 16):$ h(t)=t\,[u(t)-u(t-3)],\;x(t)=u(t)-u(t-3) $→ piecewise-quadratic output (by repeated sliding-integration).</li></ul><h3 id="32906fb1-fe4d-4325-b65d-dececdf08ce1" data-toc-id="32906fb1-fe4d-4325-b65d-dececdf08ce1" collapsed="false" seolevelmigrated="true">Algebraic Properties of Convolution</h3><ul><li>Commutative:$ xh=hx. $</li><li>Associative:$ (xh1)h2 = x(h1h2). $</li><li>Distributive:$ x(h1+h2)=xh1+x*h2. $</li><li>System interpretation:<ul><li>Associative ↔ series interconnection of LTI blocks.</li><li>Distributive ↔ parallel interconnection.</li></ul></li></ul><h3 id="09e4a405-928d-4991-b97f-3f76c9150505" data-toc-id="09e4a405-928d-4991-b97f-3f76c9150505" collapsed="false" seolevelmigrated="true">System Properties via Impulse Response</h3><ul><li>Memoryless<ul><li>DT:$ h[n]=a\,\delta[n]. $</li><li>CT:$ h(t)=a\,\delta(t). $</li></ul></li><li>Causality<ul><li>DT:$ h[n]=0\;\forall n<0. $</li><li>CT:$ h(t)=0\;\forall t<0. $</li></ul></li><li>Invertibility<ul><li>DT:$ \exists\,g[n]\;:\;g*h=\delta[n]. $</li><li>CT:$ \exists\,g(t)\;:\;g*h=\delta(t). $</li></ul></li><li>BIBO Stability<ul><li>DT:$ \sum_{n=-\infty}^{\infty}|h[n]|<\infty. $</li><li>CT:$ \int_{-\infty}^{\infty}|h(t)|\,dt<\infty. $</li></ul></li></ul><h4 id="58e19008-3a16-483a-a468-11b00df32603" data-toc-id="58e19008-3a16-483a-a468-11b00df32603" collapsed="false" seolevelmigrated="true">Practice Question (p. 24)</h4><ul><li>$ h[n]=n\Big(\tfrac13\Big)^{n}\,u[n-1] $.<ul><li>Causal? Support starts at$ n=1 $⇒ causal ✔.</li><li>Stability? Geometric series with extra$ n $:$ \sum_{n=1}^{\infty}n\,(\tfrac13)^n = \frac{\frac13}{(1-\frac13)^2}=\frac{1/3}{(2/3)^2}=\frac{1/3}{4/9}=\frac{3}{4}<\infty. $⇒ BIBO stable.</li></ul></li><li>Correct choice: (a) “causal and stable”.</li></ul><h3 id="75e51842-fde0-4707-b921-c883dcd32b6f" data-toc-id="75e51842-fde0-4707-b921-c883dcd32b6f" collapsed="false" seolevelmigrated="true">Convolution Comic Relief (p. 25)</h3><ul><li>Slides jokingly protest: “CONVOLUTION STINKS”, “MAKE LOVE NOT CONVOLUTION”, etc.</li><li>Take-away: convolution feels scary but manageable once methodical.</li></ul><h3 id="d5bbdc7f-70fe-4559-b853-4f46a72fdd64" data-toc-id="d5bbdc7f-70fe-4559-b853-4f46a72fdd64" collapsed="false" seolevelmigrated="true">Key Idea: Eigenfunction Expansion</h3><ul><li>If we possess a complete basis$ {vk(t)} $of eigenfunctions satisfying$ h*vk=\lambdak vk, $ any input$ x(t)=\sumk ak vk $produces output$ y(t)=\sumk ak \lambdak v_k. $</li><li>Greatly simplifies analysis (diagonalises the system).</li></ul><h3 id="3dccc777-dade-4f3b-ad53-c9a1e90813da" data-toc-id="3dccc777-dade-4f3b-ad53-c9a1e90813da" collapsed="false" seolevelmigrated="true">Eigenfunctions of LTI Systems</h3><ul><li>Constant input<ul><li>$ v(t)=1 $⇒$ y(t)=\Big(\int h(\tau)d\tau\Big)\cdot1, $hence eigenvalue$ \lambda=\int h(\tau)d\tau. $</li></ul></li><li>Complex exponentials<ul><li>Assume$ v(t)=e^{st},\;s\in\mathbb C. $</li><li>Output $ y(t)=e^{st}\int_{-\infty}^{\infty}h(\tau)e^{-s\tau}d\tau=H(s)e^{st}. $</li><li>$ H(s)=\int_{-\infty}^{\infty}h(\tau)e^{-s\tau}d\tau $is the transfer function = Laplace Transform of$ h $.</li><li>Therefore,$ e^{st} $is an eigenfunction with eigenvalue$ H(s). $</li></ul></li><li>Exists for all$ s $in the ROC where$ H(s) $converges.</li></ul><h3 id="d41bfa01-9b25-4a64-899a-5c53071645c1" data-toc-id="d41bfa01-9b25-4a64-899a-5c53071645c1" collapsed="false" seolevelmigrated="true">Example: Continuous-Time Integrator</h3><ul><li>System:$ y(t)=\int_{-\infty}^{t}x(\tau)\,d\tau $with impulse response$ h(t)=u(t). $</li><li>Transfer function $ H(s)=\int_{0}^{\infty}e^{-s\tau}\,d\tau=\frac{1}{s},\quad\Re{s}>0. $</li><li>Input$ x(t)=e^{t}+e^{4t} $<ul><li>Components$ e^{t}\;(s=-1)\;,e^{4t}\;(s=-4) $(note sign convention of Laplace:$ e^{st} $with$ s $positive when decaying).</li><li>Output$ y(t)=H(-1)e^{t}+H(-4)e^{4t}=\frac{1}{-1}e^{t}+\frac{1}{-4}e^{4t}=-e^{t}-\tfrac14 e^{4t}+C, $where constant of integration chosen 0 for causal integrator.</li></ul></li></ul><h3 id="bff9fe16-12d9-4baa-a632-9afb6a2b04ff" data-toc-id="bff9fe16-12d9-4baa-a632-9afb6a2b04ff" collapsed="false" seolevelmigrated="true">Summary Pointers</h3><ul><li>TI + linearity ⇒ convolution w/ impulse response.</li><li>Convolution possesses C.A.D. (commutative, associative, distributive) properties enabling modular system design.</li><li>System attributes (memory, causality, stability, invertibility) extracted solely from$ h$$.
4. Exponentials diagonalise LTI systems → frequency-domain/transfer-function analysis.
5. Practical computation: graphical folding-shifting (DT/CT) or algebraic via transforms.