Signals & Systems

• Signal - function dependent on 1+ arguments, usually describing a physical aspect of something

  • One-dimensional signal - 1 argument

  • Multi-dimensional signal - multiple arguments

  • DT signal - sequence

  • CT signal - waveform

  • Sampling turns a CT into a DT (analog to digital)

    • Sample every T (sampling interval). $y[n]=y(nT)$

      • Sampling rate 1/T
  • Reconstruction turns a DT into a CT (digital to analog)

• System - takes a signal and outputs a new signal

  • $y[n]=H\{x[n]\}$

    $y(t)=H\{x(t)\}$

• Properties of a signal, denoted x(t)

  • Energy $E=\int_{-\infty}^\infty |x(t)|^2 dt$ derivation

  • Power $P=\underset{\omega=\infty}{\lim} \frac1{2\omega} \int_{-\omega}^\omega |x(t)|^2 dt$

    • Power is average energy

    • When periodic, $P=\frac{1}{T_0}\int_{t_1}^{t_1+T_0}|x(t)|^2~dt$ for any $t_1$

  • Random - involves probability

    Deterministic

• Properties of a system, denoted H

  • Stability

    • BIBO stable (bounded-input bounded-output stable) - iff $\int_{-\infty}^{\infty}|h(t)|dt\lt\infty$

  • Memory

    • Memoryless if y[t] is only based on x[t]

    • Has memory if y[t] based on x[not t] (such as x[t-1])

  • Invertibility

    • $H$ is invertible iff there exists system $H^{-1}$ such that if $y(t)=H\{x(t)\}$ then $x(t)=H^{-1}\{y(t)\}$

  • Time-invariance

    • Time-invariant iff t-shift then apply H is the same as apply H then t-shift. Time-variance means that it depends on absolute time.

      • t-shift then apply H: $H\{x(t-t_0)\}$

      • apply H then t-shift: $y(t-t_0)$

      • Strategy when checking if time-invariant: write $x'(t)=x(t-t_0)$ to apply $H\{x'(t)\}$ and first find $y(t)$ then t-shift by $t_0$

    • Time-variant - H applies a change to x(t) differently based on each absolute t

    • Common examples

      • Variant - depends on absolute time

        • $y(t)=x(ct)$

        • $y(t)=x(c)$

        • $y(t)=tx(t)$

      • Invariant

        • y(t)=x(t-5)

  • Linearity

    • Linear iff $H\{a_1x_1(t)+a_2x_2(t)\}==a_1H\{x_1(t)\}+a_2H\{x_2(t)\}$

  • Causality

    • Causal iff y(t) dependent on only present and past values of x, that is $x(t')~~\forall ~t'\le t$

  • Real

    • A real system maps real inputs to real outputs

• Singularity functions is either $\infty$ or has no derivative at a point

• Example signals

  • Impulse function $\delta(t)$

    • CT impulse function is also called the Dirac delta function (see explanatory video here)

    • Unit doublet function $\delta'(t)=\frac{d\delta}{dt}(t)$

  • Step function $u(t)$ is 1 if $t\geq0$ else 0

  • Ramp function $r(t)=\begin{cases} t&t\geq0\\0&\text{otherwise}\end{cases}$

  • Rectangular pulse $\Pi(t) = \begin{cases} 1 & |t|\leq\frac12\\0&\text{otherwise} \end{cases}$

• Example systems

  • Modulation

    • $\underbrace{y(t)}_{\text{modulated signal}}=\underbrace{x(t)}_{\text{message signal}}~~\underbrace{\cos(\omega_ct)}_{\text{carrier signal}}$

      • $\omega_c$ is the carrier frequency

  • High-pass and low-pass filters

    • DT

      • Difference equation: $y[n] = x[n]+ay[n-1]$

      • When $0\lt a\lt 1$, low-pass filter

        When $a == 1$, running sum

        When $a \gt 1$, compound interest

        When $-1 \lt a \lt 0$, high-pass filter

      • Impulse response $h[n]=a^nu[n]$

derivation

• $\displaystyle y[n]=\sum_{k=-\infty}^n a^{n-k}x[k]$

derivation

• CT low-pass filter

  • Differential equation: $\tau y'(t) + y(t) = x(t)$ derivation

  • $y(t)=\frac1{\tau}\int_{-\infty}^t e^{-\frac{t-t'}\tau} x(t') dt'$

  • Impulse response $h(t)=\frac1\tau e^{-\frac t\tau} u(t)$

derivation

• Frequency response $H(j\omega)=\frac{1}{1+j\omega\tau}$

derivation

• -3 dB is when the power is down by 1/2 which would correspond to $|H(j\omega)|$ being $\frac1{\sqrt{2}}$ which would correspond to $1=\omega\tau$ or $\omega=\frac1\tau$ so cutoff frequency $\omega_c=\frac1\tau$

• CT high-pass filter

  • Differential equation: $\tau y'+y=\tau x'$

derivation

• $\displaystyle y(t)=\frac1{\tau}\int_{-\infty}^t e^{-\frac{t-t'}\tau} \textcolor{blue}{x'(t')} dt'$

• Impulse response $h(t)=\delta(t)-\frac1\tau e^{-\frac t\tau}u(t)$

• Frequency response $H(j\omega)=\frac{j\omega\tau}{1+j\omega\tau}$

derivation

• Squarer

  • DT

    • $y[n]=x^2[n]$

  • CT

    • $y(t) = x^2(t)$

• Initial rest - $y[n]=0$ until $x[n]$ is non-zero

• Fundamental frequency $\omega_0$

  Frequency $\omega=k\omega_0$

  • $\omega_k$ is a harmonic when $k$ is an integer

    • $\omega_1$ is the first harmonic (the fundamental frequency)

    • $\omega_2$ is the second harmonic - twice the fundamental frequency

• Convolutions

  • Convolving two signals—a(t) and b(t)—returns another signal c(t)

  • CT $c(t) = a(t) * b(t) = \int_{k=-\infty}^{\infty} a(k)b(t-k)dk$

  • DT $c[n] = a[n] * b[n] = \sum_{k=-\infty}^{\infty} a[k]b[n-k]$

  • Convolution is linear derivation

  • Convolution is commutative

derivation

• Given an LTI system H

  • Impulse response $h(t)\triangleq H\{\delta(t)\}$

    • $y[n] = x[n] * h[n]$

derivation

• Step response $s(t)\triangleq H\{u(t)\}$

• How to calculate convolution

  • Method 1: Flip and drag

    • Plot $h[k]$

    • Plot $x[n-k]$ (which is $x[-(k-n)]$, a horizontal flip and then transform right by n)

    • For each fixed value of n, take each k and multiply the two graphs, adding up all these terms to get $y[n]$

  • Method 2: Express rectangles as singularity functions and apply special functions

  • Method 3: symbolically

• Useful examples

  • $\displaystyle x[n] * u[n] = \sum_{k=-\infty}^n x[k]$ running sum

    • convolving with the step function gives a running sum

derivation

• $x[n] * \delta[n] = x[n]$ identity

  • convolving with the impulse function returns the identity

derivation

• $x[n] * \delta[n-n_0] = x[n-n_0]$ delay

derivation

• $x(t)*\delta'(t)=x'(t)$ derivative

derivation

• $\int_{-\infty}^\infty y(t)dt = \int_{-\infty}^\infty x(t)dt \int_{-\infty}^\infty h(t)dt$

derivation

• This is a useful property to check that you did your convolution correctly

• Frequency response $H(j\omega)=\displaystyle\int_{-\infty}^\infty h(t)e^{-j\omega t}dt$

derivation

• AKA transfer function $H(s)=\int_{-\infty}^\infty h(t')e^{-st'}dt'$

• If $x(t)=e^{st}$ then $y(t)=H(s)e^{st}$

derivation

• How to calculate: through the equation given above with the impulse response or by substituting in $y(t)$ as $H(s)e^{st}$ and $x(t)$ with $e^{st}$

• Properties

  • If impulse response $h(t)$ is real,

    • $H^*(j\omega)=H(-j\omega)$

derivation

• Any real $x(t)$ will yield a real $y(t)$

• Properties

  • Sampling property $x(t)\delta(t-t_0)=x(t_0)\delta(t-t_0)$

    • mn you sample that point

  • Sifting property

    • DT $\sum_{k=-\infty}^{\infty}x[k]\delta[n-k]=x[n]$

    • CT $\int_{t'=-\infty}^\infty x(t')\delta(t-t')dt'=x(t)$

    • mn sifting through the points for the one point (like sifting through water for that one piece of gold)

  • Finite

• Linear, constant-coefficient differential (CT)/difference (DT) equation

  • DT $\sum_{k=0}^N a_k y[n-k]=\sum_{k=0}^Mb_kx[n-k]$

  • CT $\sum_{k=0}^N a_k \frac{d^ky}{dt^k}(t)=\sum_{k=0}^Mb_k\frac{d^kx}{dt^k}(t)$

• Signal transforms (frequency-domain transforms)

  • Fourier series - for periodic signals (have power)

    • Fourier series - CTFS

    • Discrete Fourier series - DTFS

  • Fourier transform - for aperiodic signals (have energy)

    • Fourier transform - CTFT

    • Discrete Fourier transform - DTFT

  • Laplace transform - CT

  • Z-Transform - DT

• Fourier series

  • Basis function $\phi_k(t) = e^{jk\omega_0t}$

  • Synthesis - representing a signal $x(t)$ by a Fourier series $\hat x(t)$

    • $\displaystyle \hat x(t) = \sum_{k=-\infty}^{\infty} a_k e^{jk\omega_0 t}= \sum_{k=-\infty}^{\infty} a_k \phi_k(t)$

  • Analysis - finding the Fourier series coefficients $a_k$ to represent a signal $x(t)$

    • $\displaystyle a_k=\frac1{T_0} \int_{T_0} x(t') e^{-jk\omega_0t'}dt'=\frac1{T_0} \int_{T_0} x(t') \phi_k^*(t')dt'$

derivation

• mn $a_kT_0$ is the product of inner product of $e^{jk\omega_0t}$ and $\hat x(t)$ (approximation of x(t) with linear combo of exponentials) over one period, which is the RHS.

  mn $\int_{T_0} \hat x(t') e^{-jk\omega_0t'}dt'=\int_{T_0} x(t') e^{-jk\omega_0t'}dt'$

  LHS yields $\displaystyle \int_{T_0} \sum_{k'=-\infty}^{\infty} a_ke^{jk'\omega_0} e^{-jk\omega_0t'}dt'=a_kT_0$

• Examples

  • Pulse train: $a_k=\frac{\omega_0T_1}\pi \operatorname{sinc}(\frac{k\omega_0T_1}\pi)$

derivation

• DT pulse train $a_k=\frac1N\frac{\sin(k2\pi (N_1+\frac12)/N)}{\sin(k\pi /N)}$ for period $N$ and pulse width $2N_1$

derivation

• DT version

  • Basis function $\phi_k[n] = e^{ jk\frac{2\pi}{N} n}$

  • Synthesis $\displaystyle \hat x[n]=\sum_{k=\langle N\rangle}a_ke^{ jk\frac{2\pi}{N} n}$

  • Analysis $\displaystyle a_k=\frac1N\sum_{n=\langle N\rangle}x[n]e^{-jk\frac{2\pi}Nn}$

  • Fundamental frequency $\Omega_0=\frac{2\pi}N$

• $\operatorname{sinc}(x)=\frac{\sin \pi x}{\pi x}$

• With LTI system H

  • If $y(t)=H\{x(t)\}=H(j\omega)x(t)$ and $x(t)\stackrel{\rm FS}\leftrightarrow a_k$ then $y(t)\stackrel{\rm FS}\leftrightarrow a_kH(j\omega)$ due to linearity of H

• Properties given $x(t)\stackrel {\rm FS}\leftrightarrow a_k$ and $y(t)\stackrel{\rm FS}\leftrightarrow b_k$

  • Both

    • Linearity $Ax(t)+By(t)\leftrightarrow Aa_k+Bb_k$

derivation

• Power $\displaystyle P=\frac1{T_0}\int_{T_0}|x(t)|^2dt=\sum_{k=-\infty}^\infty|a_k|^2$ derivation

• If $x(t)$ is real, $a_{-k}=a_{k}^*$

derivation

• If $x(t)$ is even and real, $a_k$ is even and real. If $x(t)$ is odd and real, $a_k$ is odd and imaginary

derivation

• T-shift delay: $x(t-t_0)\leftrightarrow a_k\textcolor{blue}{e^{-jk\omega_0t_0}}$ if $x(t)\leftrightarrow a_k$

derivation

• CTFS

  • Parseval's identity $\displaystyle \langle x(t),y(t)\rangle = \int_{T_0} x(t)y^*(t)dt=T_0\sum_{k=-\infty}^\infty a_kb^*_k$ - relates the inner product of two signals to their FS coefficients derivation

• Group delay $-\frac{d\angle H(j\omega)}{d\omega}$ - how delayed (phase-shifted) the components are based on their frequency

• Continuous-time Fourier transform (CTFT)

  • CTFT $\displaystyle X(j\omega)=\int_{-\infty}^\infty x(t) e^{-j\omega t}dt$

    • AKA analysis equation ∵ analysis is extracting coeffs $\displaystyle X(j\omega)$

    • $a_k=\frac1{T_0}X(jk\omega_0)$

      • where $X(j\omega)$ is for aperiodic signal version (one period) and $a_k$ is for periodic signal (CTFS coeff)

  • Inverse CTFT $\displaystyle x(t)=\frac{1}{2\pi} \int_{-\infty}^\infty X(j\omega) e^{j\omega t}d\omega$

derivation

• AKA CTFT synthesis equation ∵ synthesis is synthesizing signal from coeffs

• mn synthesis has +$j\omega t$ because synthesis is about creating (positive) whereas analysis is about taking apart (negative) so it has a -$j\omega t$

• This means that for frequency response $H(j\omega)$, $\displaystyle H\{x(t)\}=\frac{1}{2\pi} \int_{-\infty}^\infty X(j\omega) H(j\omega) e^{j\omega t}d\omega$

  $\displaystyle =\frac{1}{2\pi} \int_{-\infty}^\infty |X(j\omega)|e^{\langle X(j\omega)} |H(j\omega)|e^{\langle H(j\omega)} e^{j\omega t}d\omega$

  $\displaystyle =\frac{1}{2\pi} \int_{-\infty}^\infty |X(j\omega)H(j\omega)|e^{\langle X(j\omega)+\langle H(j\omega)} e^{j\omega t}d\omega$

  so the phase distortion $\angle H(j\omega)$ tells you how much the frequencies are delayed

• Notation

  • CTFT operator $\operatorname F$: $F[x(t)]=X(j\omega)$ and $F^{-1}[X(j\omega)] = x(t)$

  • CTFT pair: $x(t) \stackrel {F}\leftrightarrow X(j\omega)$

• Eigenfunction of CT LTI system is $e^{jk\omega_0t}$

derivation

• $Y(j\omega)=X(j\omega)H(j\omega)$

derivation

• If the Dirichlet conditions (below) are satisfied, then synthesized signal $\hat x(t)=x(t)$ except near $t$ when $x(t)$ has discontinuities and the energy difference between them vanishes: $\int_{-\infty}^\infty |x(t)-\hat x(t)|^2 dt=0$

  • $x(t)$ is absolutely integrable $\int_{-\infty}^\infty |x(t)|dt <\infty$

  • $x(t$) has a finite number of local minima and maxima in a finite interval

  • $x(t)$ has a finite number of discontinuities in a finite interval

  • $x(t)$'s discontinuities are all finite

• Examples

  • $F\{ \Pi(\frac t {2T_1}) \} = 2T_1\operatorname{sinc} \frac{\omega T_1}\pi$

derivation

• $F\{1\}=2\pi\delta(\omega)$

derivation

• $F\{\delta(t)\} = 1$

derivation

• $F\{\frac W \pi \operatorname{sinc} (\frac{Wt}\pi)\}=\Pi(\frac \omega {2W})$

derivation

• $F\{e^{j\omega_0t}\}=2\pi\delta(\omega-\omega_0)$

derivation

• From CTFS: $\displaystyle F\{x(t)\}=F\left\{\sum_{k=-\infty}^\infty a_k e^{jk\omega_0 t}\right\}=2\pi\sum_{k=-\infty}^\infty a_k\delta(\omega-k\omega_0)$

derivation

• Impulse train: $\displaystyle F\left\{\sum_{n=-\infty}^\infty\delta(t-nT_0)\right\}=\frac{2\pi}{T_0}\sum_{k=-\infty}^\infty\delta(\omega-k\omega_0)$

derivation

• $F\{u(t)\}=\frac1{j\omega}+\pi\delta(\omega)$

derivation

• $F\{\cos(\omega_0 t)\} = \pi(\delta(\omega-\omega_0)+\delta(\omega+\omega_0))$

derivation

• $F\{\sin(\omega_0t)\}=\frac{\pi}j(\delta(\omega-\omega_0)-\delta(\omega+\omega_0))$

derivation

• $F\{\operatorname{sgn}(t)\}=\frac2{j\omega}$

• Properties

  • Linearity: $Ax(t)+By(t)\overset F\leftrightarrow AX(j\omega)+BY(j\omega)$

derivation

• Time scaling: $x(at) \overset F \leftrightarrow \frac1{|a|}X(j\frac\omega a)$

derivation

• Corollary: $F\{x(-t)\} = X(-j\omega)$

derivation

• Time shift $F\{x(t-t_0)\}=F\{x(t)\}e^{-j\omega t_0}$

derivation

• $F\{x^*(t)\}=X^*(-j\omega)$

derivation

• Corollary: if a signal x(t) is real, its CTFT has conjugate symmetry

• If $x(t)$ is real and even, $X(j\omega)$ is real and even

  If $x(t)$ is real and odd, $X(j\omega)$ is imaginary and odd

derivation

• $F\{\frac{dx}{dt}\}=j\omega X(j\omega)$ (differentiation)

derivation

• $F\{x(t)e^{j\omega_0t}\}=X(j(\omega-\omega_0))$ (frequency shifting)

derivation

• $\displaystyle F\left\{ \int_{-\infty}^t x(t') dt' \right\}=\frac1{j\omega}X(j\omega)+\pi X(j0)\delta(\omega)$

• $F\{tx(t)\}=j\frac{dX}{d\omega}(j\omega)$

derivation

• Parseval's identity: $\displaystyle \langle x(t),y(t)\rangle=\frac1{2\pi}\int_{-\infty}^\infty X(j\omega)Y^*(j\omega)d\omega$, which is $\displaystyle \int_{-\infty}^\infty x(t)y^*(t)dt$

derivation

• Energy of x(t) is $\displaystyle \frac1{2\pi}\int_{-\infty}^\infty |X(j\omega)|^2 d\omega$, which is also $\int_{-\infty}^\infty |x(t)|^2 dt=\langle x(t), x(t) \rangle$

derivation

• $F\{x(t)*y(t)\}=X(j\omega)Y(j\omega)$

derivation

• $\displaystyle F\{x(t)y(t)\}=\frac1{2\pi}X(j\omega)*Y(j\omega)$

derivation

• DTFS

  • $\displaystyle X(e^{j\Omega})=\sum_{n=-\infty}^\infty x[n]e^{-j\Omega n}$ - analysis

    • $a_k=\frac1NX(e^{jk\Omega_0})$

  • $\displaystyle x[n] = \sum_{k=\langle N \rangle} a_k e^{jk\Omega_0 n} \overset{F}{\leftrightarrow} X(e^{j\Omega}) = 2\pi \sum_{k=-\infty}^{\infty} a_k \delta(\Omega - k\Omega_0)$

  • Eigenfunction of DT LTI system is $z^n$. When $z=e^{j\Omega}$, we can use $H\{e^{j\Omega}\}=e^{j\Omega n}$

derivation

• $Y(j\omega)=X(j\Omega)H(j\Omega)$

derivation

• DTFT

  • 

Additional Notes

• Conjugate symmetry: $f(x)=f(-x)^*$ means $f$ has conjugate symmetry

• Helpful math

  • $e^{j\theta}+e^{-j\theta}=2\cos\theta$

  • $e^{j\theta}-e^{-j\theta}=2j\sin\theta$

derivation

• Conjugation distributes: $(ab)^*=a^*b^*$

derivation

• Inner product $\langle x(t), y(t) \rangle \stackrel\Delta= \int_{T_0} x(t)y^*(t)dt$

  • CT version of dot product (DT)

• $\displaystyle \int_{-T}^T e^{j\omega t}dt = 2T\operatorname{sinc}\frac{\omega T}\pi$

derivation