Systems Properties & LTI Convolution

Time–Invariant (TI) Systems

  • Definition

    • For every admissible input x(t)  (or x[n])x(t)\;(\text{or }x[n]) and every time-shift t<em>0  (or n</em>0)t<em>0\;(\text{or }n</em>0),
      System input x(t−t<em>0)  →  output y(t−t</em>0)\text{System input }x(t-t<em>0)\;\to\;\text{output }y(t-t</em>0)
      (analogously in DT).

  • Formal condition

    • CT: x(t−t<em>0)→sysy(t−t</em>0)  ∀ t0∈Rx(t-t<em>0)\xrightarrow{\text{sys}}y(t-t</em>0)\;\forall\,t_0\in\mathbb R

    • DT: x[n−n<em>0]→sysy[n−n</em>0]  ∀ n0∈Zx[n-n<em>0]\xrightarrow{\text{sys}}y[n-n</em>0]\;\forall\,n_0\in\mathbb Z

  • Test examples

    • y(t)=sin⁡(x(t))y(t)=\sin(x(t)) → TI ✔ (time enters neither explicitly nor via coefficients).

    • y[n]=n x[n]y[n]=n\,x[n] → TI ✘ (explicit dependence on $n$ violates invariance).

    • y[n]=x[n]x[n−3]y[n]=x[n]x[n-3] → TI ✔ (all time indices shifted identically).

    • y(t)=x(5t)y(t)=x(5t) → TI ✘ (time scaling is not a pure shift).

Linear Systems

  • Definition (superposition)

    • For any inputs x<em>1,x</em>2x<em>1,x</em>2 and scalars a<em>1,a</em>2∈Ca<em>1,a</em>2\in\mathbb C
      x(t)=a<em>1x</em>1(t)+a<em>2x</em>2(t)  ⇒  y(t)=a<em>1y</em>1(t)+a<em>2y</em>2(t).x(t)=a<em>1x</em>1(t)+a<em>2x</em>2(t)\;\Rightarrow\;y(t)=a<em>1y</em>1(t)+a<em>2y</em>2(t).

  • Test examples

    • Integrator y(t)=∫1tx(τ) dτy(t)=\int_{1}^{t}x(\tau)\,d\tau – Linear ✔ (limits independent of input; additive & homogeneous).

    • y[n]=(x[2n])2y[n]=(x[2n])^2 – Linear ✘ (square breaks homogeneity).

    • y(t)=(t−1)x(t+2)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 δ[n]\delta[n].

  • Impulse response: h[n]=output to δ[n].h[n]=\text{output to }\delta[n].

  • Any sequence can be decomposed:
    x[n]=∑k=−∞∞x[k] δ[n−k].x[n]=\sum_{k=-\infty}^{\infty}x[k]\,\delta[n-k].

Convolution sum

  • Via TI + linearity:
    y[n]=∑k=−∞∞x[k]h[n−k]  ≜  (x∗h)[n].y[n]=\sum_{k=-\infty}^{\infty}x[k]h[n-k]\;\triangleq\;(x*h)[n].

  • Commutative form: ∑<em>kx[k]h[n−k]=∑</em>kx[n−k]h[k].\sum<em>{k}x[k]h[n-k]=\sum</em>{k}x[n-k]h[k].

Illustrative example (p. 5)

  • Given h[n]=u[n]h[n]=u[n] (sketched as 0.2,0.4,0.6,0.8,1) and input x[n]=u[n]−u[n−3]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)≈x^(t)=∑<em>k=−∞∞x(kΔ) p</em>Δ(t−kΔ).x(t)\approx\hat x(t)=\sum<em>{k=-\infty}^{\infty}x(k\Delta)\,p</em>{\Delta}(t-k\Delta).

  • Let h<em>Δ(t)=response to p</em>Δ(t).h<em>{\Delta}(t)=\text{response to }p</em>{\Delta}(t).

  • By TI+linearity: y^(t)=∑<em>kx(kΔ)h</em>Δ(t−kΔ).\hat y(t)=\sum<em>k x(k\Delta)h</em>{\Delta}(t-k\Delta).

  • As Δ→0\Delta\to0

    • p<em>Δ(t)→δ(t)p<em>{\Delta}(t)\to\delta(t), h</em>Δ(t)→h(t).h</em>{\Delta}(t)\to h(t).

    • Riemann sum → integral ⇒
      y(t)=∫−∞∞x(τ)h(t−τ) dτ  ≜  (x∗h)(t).y(t)=\int_{-\infty}^{\infty}x(\tau)h(t-\tau)\,d\tau\;\triangleq\;(x*h)(t).

Impulse response definition

  • h(t)=output to δ(t).h(t)=\text{output to }\delta(t).

  • Convolution integral as above.

Main Message

  • Knowing h(t)  (or h[n])h(t)\;(\text{or }h[n]) fully characterises an LTI system:
    h  ◯  x  ⟶  y.h \;\bigcirc\; x \;\longrightarrow\; y.

  • Challenge: direct convolution can be algebraically heavy.

Identifying LTI Systems (p. 10)

  • a) y[n]=3x[n]+2y[n]=3x[n]+2 – Not LTI (fails linearity: constant term).

  • b) y(t)=12π∫−∞∞x(τ) [u(t−τ)−u(t−τ−1)] dτy(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)y(t)=x(2t) – TI ✘, so not LTI.

  • d) y[n]=(x[2n])2y[n]=(x[2n])^2 – Non-linear.

Computing Convolution (DT)

  • Algorithm (fix $n$):

    1. Flip h[k]→h[−k]h[k]\to h[-k].

    2. Shift by $n\;\to h[n-k]$.

    3. Multiply with x[k] sample-wise.

    4. Sum over $k$.

  • Example (p. 11–15):

    • h[n]=n(u[n]-u[n-3]),\;x[n]=u[n]-u[n-3].</p></li><li><p>Non−zerosupport0–2.Convolutionyieldstriangularshapeoflength5,numerically:</p></li><li><p>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.</p></li></ul></li></ul><h3id="4ddb6529−27b9−47ad−8fff−f34fd9bb5098"data−toc−id="4ddb6529−27b9−47ad−8fff−f34fd9bb5098"collapsed="false"seolevelmigrated="true">ComputingConvolution(CT)</h3><ul><li><p>Formula</p></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><p>Formulay(t)=\int_{-\infty}^{\infty}x(\tau)h(t-\tau)\,d\tau.

    • Procedure mirrors DT: flip $h$, shift to $t$, multiply with $x$, integrate.

    • Example (p. 16): h(t)=t\,[u(t)-u(t-3)],\;x(t)=u(t)-u(t-3)→piecewise−quadraticoutput(byrepeatedsliding−integration).</p></li></ul><h3id="32906fb1−fe4d−4325−b65d−dececdf08ce1"data−toc−id="32906fb1−fe4d−4325−b65d−dececdf08ce1"collapsed="false"seolevelmigrated="true">AlgebraicPropertiesofConvolution</h3><ul><li><p>Commutative:→ piecewise-quadratic output (by repeated sliding-integration).</p></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><p>Commutative:xh=hx.</p></li><li><p>Associative:</p></li><li><p>Associative:(xh1)h2 = x(h1h2).</p></li><li><p>Distributive:</p></li><li><p>Distributive:x(h1+h2)=xh1+x*h2.</p></li><li><p>Systeminterpretation:</p><ul><li><p>Associative↔seriesinterconnectionofLTIblocks.</p></li><li><p>Distributive↔parallelinterconnection.</p></li></ul></li></ul><h3id="09e4a405−928d−4991−b97f−3f76c9150505"data−toc−id="09e4a405−928d−4991−b97f−3f76c9150505"collapsed="false"seolevelmigrated="true">SystemPropertiesviaImpulseResponse</h3><ul><li><p>Memoryless</p><ul><li><p>DT:</p></li><li><p>System interpretation:</p><ul><li><p>Associative ↔ series interconnection of LTI blocks.</p></li><li><p>Distributive ↔ parallel interconnection.</p></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><p>Memoryless</p><ul><li><p>DT:h[n]=a\,\delta[n].</p></li><li><p>CT:</p></li><li><p>CT:h(t)=a\,\delta(t).</p></li></ul></li><li><p>Causality</p><ul><li><p>DT:</p></li></ul></li><li><p>Causality</p><ul><li><p>DT:h[n]=0\;\forall n<0.</p></li><li><p>CT:</p></li><li><p>CT:h(t)=0\;\forall t<0.</p></li></ul></li><li><p>Invertibility</p><ul><li><p>DT:</p></li></ul></li><li><p>Invertibility</p><ul><li><p>DT:\exists\,g[n]\;:\;g*h=\delta[n].</p></li><li><p>CT:</p></li><li><p>CT:\exists\,g(t)\;:\;g*h=\delta(t).</p></li></ul></li><li><p>BIBOStability</p><ul><li><p>DT:</p></li></ul></li><li><p>BIBO Stability</p><ul><li><p>DT:\sum_{n=-\infty}^{\infty}|h[n]|<\infty.</p></li><li><p>CT:</p></li><li><p>CT:\int_{-\infty}^{\infty}|h(t)|\,dt<\infty.</p></li></ul></li></ul><h4id="58e19008−3a16−483a−a468−11b00df32603"data−toc−id="58e19008−3a16−483a−a468−11b00df32603"collapsed="false"seolevelmigrated="true">PracticeQuestion(p.24)</h4><ul><li><p></p></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><p>h[n]=n\Big(\tfrac13\Big)^{n}\,u[n-1].

      • Causal? Support starts at $n=1$ ⇒ causal ✔.

      • 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.⇒BIBOstable.</p></li></ul></li><li><p>Correctchoice:(a)“causalandstable”.</p></li></ul><h3id="75e51842−fde0−4707−b921−c883dcd32b6f"data−toc−id="75e51842−fde0−4707−b921−c883dcd32b6f"collapsed="false"seolevelmigrated="true">ConvolutionComicRelief(p.25)</h3><ul><li><p>Slidesjokinglyprotest:“CONVOLUTIONSTINKS”,“MAKELOVENOTCONVOLUTION”,etc.</p></li><li><p>Take−away:convolutionfeelsscarybutmanageableoncemethodical.</p></li></ul><h3id="d5bbdc7f−70fe−4559−b853−4f46a72fdd64"data−toc−id="d5bbdc7f−70fe−4559−b853−4f46a72fdd64"collapsed="false"seolevelmigrated="true">KeyIdea:EigenfunctionExpansion</h3><ul><li><p>Ifwepossessacompletebasis⇒ BIBO stable.</p></li></ul></li><li><p>Correct choice: (a) “causal and stable”.</p></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><p>Slides jokingly protest: “CONVOLUTION STINKS”, “MAKE LOVE NOT CONVOLUTION”, etc.</p></li><li><p>Take-away: convolution feels scary but manageable once methodical.</p></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><p>If we possess a complete basis{vk(t)}ofeigenfunctionssatisfyingof eigenfunctions satisfyingh*vk=\lambdak vk,<br>anyinput<br>any inputx(t)=\sumk ak vkproducesoutputproduces outputy(t)=\sumk ak \lambdak v_k.</p></li><li><p>Greatlysimplifiesanalysis(diagonalisesthesystem).</p></li></ul><h3id="3dccc777−dade−4f3b−ad53−c9a1e90813da"data−toc−id="3dccc777−dade−4f3b−ad53−c9a1e90813da"collapsed="false"seolevelmigrated="true">EigenfunctionsofLTISystems</h3><ul><li><p>Constantinput</p><ul><li><p></p></li><li><p>Greatly simplifies analysis (diagonalises the system).</p></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><p>Constant input</p><ul><li><p>v(t)=1⇒⇒y(t)=\Big(\int h(\tau)d\tau\Big)\cdot1,henceeigenvaluehence eigenvalue\lambda=\int h(\tau)d\tau.</p></li></ul></li><li><p>Complexexponentials</p><ul><li><p>Assume</p></li></ul></li><li><p>Complex exponentials</p><ul><li><p>Assumev(t)=e^{st},\;s\in\mathbb C.</p></li><li><p>Output<br></p></li><li><p>Output<br>y(t)=e^{st}\int_{-\infty}^{\infty}h(\tau)e^{-s\tau}d\tau=H(s)e^{st}.</p></li><li><p></p></li><li><p>H(s)=\int_{-\infty}^{\infty}h(\tau)e^{-s\tau}d\tauisthetransferfunction=LaplaceTransformofis the transfer function = Laplace Transform ofh.</p></li><li><p>Therefore,.</p></li><li><p>Therefore,e^{st}isaneigenfunctionwitheigenvalueis an eigenfunction with eigenvalueH(s).</p></li></ul></li><li><p>Existsforall</p></li></ul></li><li><p>Exists for allsintheROCwherein the ROC whereH(s)converges.</p></li></ul><h3id="d41bfa01−9b25−4a64−899a−5c53071645c1"data−toc−id="d41bfa01−9b25−4a64−899a−5c53071645c1"collapsed="false"seolevelmigrated="true">Example:Continuous−TimeIntegrator</h3><ul><li><p>System:converges.</p></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><p>System:y(t)=\int_{-\infty}^{t}x(\tau)\,d\tauwithimpulseresponsewith impulse responseh(t)=u(t).</p></li><li><p>Transferfunction<br></p></li><li><p>Transfer function<br>H(s)=\int_{0}^{\infty}e^{-s\tau}\,d\tau=\frac{1}{s},\quad\Re{s}>0.</p></li><li><p>Input</p></li><li><p>Inputx(t)=e^{t}+e^{4t}</p><ul><li><p>Components</p><ul><li><p>Componentse^{t}\;(s=-1)\;,e^{4t}\;(s=-4) (note sign convention of Laplace: $e^{st}$ with $s$ positive when decaying).

      • 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,whereconstantofintegrationchosen0forcausalintegrator.</p></li></ul></li></ul><h3id="bff9fe16−12d9−4baa−a632−9afb6a2b04ff"data−toc−id="bff9fe16−12d9−4baa−a632−9afb6a2b04ff"collapsed="false"seolevelmigrated="true">SummaryPointers</h3><ul><li><p>TI+linearity⇒convolutionw/impulseresponse.</p></li><li><p>ConvolutionpossessesC.A.D.(commutative,associative,distributive)propertiesenablingmodularsystemdesign.</p></li><li><p>Systemattributes(memory,causality,stability,invertibility)extractedsolelyfromwhere constant of integration chosen 0 for causal integrator.</p></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><p>TI + linearity ⇒ convolution w/ impulse response.</p></li><li><p>Convolution possesses C.A.D. (commutative, associative, distributive) properties enabling modular system design.</p></li><li><p>System attributes (memory, causality, stability, invertibility) extracted solely fromh$$.

      • Exponentials diagonalise LTI systems → frequency-domain/transfer-function analysis.

      • Practical computation: graphical folding-shifting (DT/CT) or algebraic via transforms.