0.15 Appendix a

Just because some of us can read and write and do a little math, that doesn't mean we deserve to conquer the Universe.

—Kurt Vonnegut, Hocus Pocus , 1990

This appendix gathers together all of the math facts used in the text. They are divided into six categories:

• Trigonometric identities
• Fourier transforms and properties
• Energy and power
• Z-transforms and properties
• Integral and derivative formulas
• Matrix algebra

So, with no motivation or interpretation, just labels, here they are:

Trigonometric identities

• Euler's relation
  $$e^{\pm jx}=\cos \left(x\right)\pm j\sin \left(x\right)$$
• Exponential definition of a cosine
  $$\cos \left(x\right)=\frac{1}{2}\left(e^{jx},+,e^{-jx}\right)$$
• Exponential definition of a sine
  $$\sin \left(x\right)=\frac{1}{2j}\left(e^{jx},-,e^{-jx}\right)$$
• Cosine squared
  $${\cos }^2\left(x\right)=\frac{1}{2}\left(1,+,\cos ,\left(,2,x,\right)\right)$$
• Sine squared
  $${\sin }^2\left(x\right)=\frac{1}{2}\left(1,-,\cos ,\left(,2,x,\right)\right)$$
• Sine and Cosine as phase shifts of each other
  $$\begin{array}{ccc}\hfill \sin \left(x\right)& =& \cos \left(\frac{\pi }{2},-,x\right)=\cos \left(x,-,\frac{\pi }{2}\right)\hfill \\ \hfill \cos \left(x\right)& =& \sin \left(\frac{\pi }{2},-,x\right)=-\sin \left(x,-,\frac{\pi }{2}\right)\hfill \end{array}$$
• Sine–cosine product
  $$\sin \left(x\right)\cos \left(y\right)=\frac{1}{2}\left[\sin ,(,x,-,y,),+,\sin ,(,x,+,y,)\right]$$
• Cosine–cosine product
  $$\cos \left(x\right)\cos \left(y\right)=\frac{1}{2}\left[\cos ,(,x,-,y,),+,\cos ,(,x,+,y,)\right]$$
• Sine–sine product
  $$\sin \left(x\right)\sin \left(y\right)=\frac{1}{2}\left[\cos ,(,x,-,y,),-,\cos ,(,x,+,y,)\right]$$
• Odd symmetry of the sine
  $$\sin (-x)=-\sin \left(x\right)$$
• Even symmetry of the cosine
  $$\cos (-x)=\cos \left(x\right)$$
• Cosine angle sum
  $$\cos (x\pm y)=\cos \left(x\right)\cos \left(y\right)\mp \sin \left(x\right)\sin \left(y\right)$$
• Sine angle sum
  $$\sin (x\pm y)=\sin \left(x\right)\cos \left(y\right)\pm \cos \left(x\right)\sin \left(y\right)$$

Fourier transforms and properties

• Definition of Fourier transform
  $$W\left(f\right)={\int }_{-\infty }^{\infty }w\left(t\right)e^{-j2\pi ft}dt$$
• Definition of Inverse Fourier transform
  $$w\left(t\right)={\int }_{-\infty }^{\infty }W\left(f\right)e^{j2\pi ft}df$$
• Fourier transform of a sine
  $$\begin{array}{ccc}\hfill \mathcal{F}& & \left\{A\sin (2\pi f_{0}t+\Phi )\right\}\hfill \\ & & =j\frac{A}{2}\left[-,e^{j\Phi },\delta ,(f-f_0),+,e^{-j\Phi },\delta ,(f+f_0)\right]\hfill \end{array}$$
• Fourier transform of a cosine
  $$\begin{array}{ccc}\hfill \mathcal{F}& & \left\{A\cos (2\pi f_{0}t+\Phi )\right\}\hfill \\ & & =\frac{A}{2}\left[e^{j\Phi },\delta ,(f-f_0),+,e^{-j\Phi },\delta ,(f+f_0)\right]\hfill \end{array}$$
• Fourier transform of impulse
  $$\mathcal{F}\left\{\delta \right(t\left)\right\}=1$$
• Fourier transform of rectangular pulse
• With
  $$\Pi \left(t\right)=\left\{\begin{array}{cc}1\hfill & -T/2\le t\le T/2\\ 0\hfill & \text{otherwise}\end{array}\right),$$
  $$\mathcal{F}\left\{\Pi \left(t\right)\right\}=T\frac{\sin \left(\pi fT\right)}{\pi fT}\equiv T\text{sinc}\left(fT\right).$$
• Fourier transform of sinc function
  $$\mathcal{F}\left\{\text{sinc}\left(2Wt\right)\right\}=\frac{1}{2W}\Pi \left(\frac{f}{2W}\right)$$
• Fourier transform of raised cosine
• With
  $$w\left(t\right)=2f_0\left(\frac{\sin \left(2\pi f_{0}t\right)}{2\pi f_{0}t}\right)\left[\frac{\cos \left(2\pi f_{\Delta }t\right)}{1-{\left(4f_{\Delta }t\right)}^2}\right],$$
  $$\mathcal{F}\left\{w\left(t\right)\right\}=\left\{\begin{array}{cc}1\hfill & \left|f\right|<f_1\hfill \\ {\displaystyle \frac{1}{2}\left(1,+,\cos ,\left[\frac{\pi \left(\right|f|-f_1)}{2f_{\Delta }}\right]\right)}\hfill & f_1<\left|f\right|<B\hfill \\ 0\hfill & \left|f\right|>B\hfill \end{array}\right),$$
  with the rolloff factor defined as $\beta =f_{\Delta }/f_0$ .
• Fourier transform of square-root raised cosine (SRRC)
• With $w\left(t\right)$ given by
  $$\left\{\begin{array}{cc}{\displaystyle \frac{1}{\sqrt{T}}\frac{\sin \left(\pi \right(1-\beta )t/T)+(4\beta t/T)\cos \left(\pi \right(1+\beta )t/T)}{(\pi t/T)(1-{(4\beta t/T)}^2)}}\hfill & t\ne 0,\pm \frac{T}{4\beta }\hfill \\ {\displaystyle \frac{1}{\sqrt{T}}(1-\beta +(4\beta /\pi ))}\hfill & t=0\hfill \\ {\displaystyle \frac{\beta }{\sqrt{2T}}\left[\left(1,+,\frac{2}{\pi }\right),\sin ,\left(\frac{\pi }{4\beta }\right),+,\left(1,-,\frac{2}{\pi }\right),\cos ,\left(\frac{\pi }{4\beta }\right)\right]}\hfill & t=\pm \frac{T}{4\beta }\hfill \end{array}\right),$$
  $$\mathcal{F}\left\{w\left(t\right)\right\}=\left\{\begin{array}{cc}1\hfill & \left|f\right|<f_1\hfill \\ {\displaystyle {\left[\frac{1}{2},\left(1,+,\cos ,\left[\frac{\pi \left(\right|f|-f_1)}{2f_{\Delta }}\right]\right)\right]}^{1/2}}\hfill & f_1<\left|f\right|<B\hfill \\ 0\hfill & \left|f\right|>B\hfill \end{array}\right).$$
• Fourier transform of periodic impulse sampled signal
• With
  $$\mathcal{F}\left\{w\right(t\left)\right\}=W\left(f\right),$$
  and
  $$\begin{array}{ccc}\hfill w_s\left(t\right)& =& w\left(t\right)\sum _{k=-\infty }^{\infty }\delta (t-k{T}_s),\hfill \\ \hfill \mathcal{F}\left\{w_s\left(t\right)\right\}& =& \frac{1}{T_s}\sum _{n=-\infty }^{\infty }W(f-(n/T_s)).\hfill \end{array}$$
• Fourier transform of a step
• With
  $$\begin{array}{ccc}\hfill w\left(t\right)& =& \left\{\begin{array}{cc}A\hfill & t>0\hfill \\ 0\hfill & t<0\hfill \end{array}\right),\hfill \\ \hfill \mathcal{F}\left\{w\right(t\left)\right\}& =& A\left[\frac{\delta \left(f\right)}{2},+,\frac{1}{j2\pi f}\right].\hfill \end{array}$$
• Fourier transform of ideal $\pi /2$ phase shifter (Hilbert transformer) filterimpulse response
• With
  $$\begin{array}{ccc}\hfill w\left(t\right)& =& \left\{\begin{array}{cc}\frac{1}{\pi t}& t>0\\ 0& t<0\end{array}\right\},\hfill \\ \hfill \mathcal{F}\left\{w\right(t\left)\right\}& =& \left\{\begin{array}{cc}-j& f>0\hfill \\ j& f<0\hfill \end{array}\right).\hfill \end{array}$$
• Linearity property
• With $\mathcal{F}\left\{w_i\left(t\right)\right\}=W_i\left(f\right)$ ,
  $$\mathcal{F}\{a{w}_1\left(t\right)+b{w}_2\left(t\right)\}=a{W}_1\left(f\right)+b{W}_2\left(f\right).$$
• Duality property With $\mathcal{F}\left\{w\right(t\left)\right\}=W\left(f\right)$ ,
  $$\mathcal{F}\left\{W\right(t\left)\right\}=w(-f).$$
• Cosine modulation frequency shift property
• With $\mathcal{F}\left\{w\right(t\left)\right\}=W\left(f\right)$ ,
  $$\begin{array}{ccc}\hfill \mathcal{F}& & \left\{w\left(t\right)\cos (2\pi f_{c}t+\theta )\right\}\hfill \\ & & =\frac{1}{2}\left[e^{j\theta },W,(f-f_c),+,e^{-j\theta },W,(f+f_c)\right].\hfill \end{array}$$
• Exponential modulation frequency shift property
• With $\mathcal{F}\left\{w\right(t\left)\right\}=W\left(f\right)$ ,
  $$\mathcal{F}\left\{w\left(t\right)e^{j2\pi f_{0}t}\right\}=W(f-f_0).$$
• Complex conjugation (symmetry) property If $w\left(t\right)$ is real valued,
  $$W^{*}\left(f\right)=W(-f),$$
  where the superscript $*$ denotes complex conjugation (i.e.,  ${(a+jb)}^{*}=a-jb)$ . In particular, $\left|W\right(f\left)\right|$ is even and $\angle W\left(f\right)$ is odd.
• Symmetry property for real signals Suppose $w\left(t\right)$ is real.
  $$\begin{array}{ccc}& \text{If}& w\left(t\right)=w(-t),\text{then}W\left(f\right)\text{is}\text{real.}\hfill \\ & \text{If}& w\left(t\right)=-w(-t),\hfill \\ & & W\left(f\right)\text{is}\text{purely}\text{imaginary.}\hfill \end{array}$$
• Time shift property
• With $\mathcal{F}\left\{w\right(t\left)\right\}=W\left(f\right)$ ,
  $$\mathcal{F}\left\{w(t-t_0)\right\}=W\left(f\right)e^{-j2\pi f{t}_0}.$$
• Frequency scale property
• With $\mathcal{F}\left\{w\right(t\left)\right\}=W\left(f\right)$ ,
  $$\mathcal{F}\left\{w\left(at\right)\right\}=\frac{1}{a}W\left(\frac{f}{a}\right).$$
• Differentiation property
• With $\mathcal{F}\left\{w\right(t\left)\right\}=W\left(f\right)$ ,
  $$\frac{dw\left(t\right)}{dt}=j2\pi fW\left(f\right).$$
• Convolution $\leftrightarrow$ multiplication property
• With $\mathcal{F}\left\{w_i\left(t\right)\right\}=W_i\left(f\right)$ ,
  $$\mathcal{F}\{w_1\left(t\right)*w_2\left(t\right)\}=W_1\left(f\right)W_2\left(f\right)$$
  and
  $$\mathcal{F}\left\{w_1\left(t\right)w_2\left(t\right)\right\}=W_1\left(f\right)*W_2\left(f\right),$$
  where the convolution operator “ $*$ ” is defined via
  $$x\left(\alpha \right)*y\left(\alpha \right)\equiv {\int }_{-\infty }^{\infty }x\left(\lambda \right)y(\alpha -\lambda )d\lambda .$$
• Parseval's theorem
• With $\mathcal{F}\left\{w_i\left(t\right)\right\}=W_i\left(f\right)$ ,
  $${\int }_{-\infty }^{\infty }{w}_1\left(t\right)w_2^{*}\left(t\right)dt={\int }_{-\infty }^{\infty }{W}_1\left(f\right)W_2^{*}\left(f\right)df.$$
• Final value theorem
• With ${lim}_{t\to -\infty }w\left(t\right)=0$ and $w\left(t\right)$ bounded,
  $$\underset{t\to \infty }{lim}w\left(t\right)=\underset{f\to 0}{lim}j2\pi fW\left(f\right),$$
  where $\mathcal{F}\left\{w\right(t\left)\right\}=W\left(f\right)$ .

Energy and power

• Energy of a continuous time signal $s\left(t\right)$ is
  $$E\left(s\right)={\int }_{-\infty }^{\infty }{s}^2\left(t\right)dt$$
  if the integral is finite.
• Power of a continuous time signal $s\left(t\right)$ is
  $$P\left(s\right)=\underset{T\to \infty }{lim}\frac{1}{T}{\int }_{-T/2}^{T/2}{s}^2\left(t\right)dt$$
  if the limit exists.
• Energy of a discrete time signal $s\left[k\right]$ is
  $$E\left(s\right)=\sum _{-\infty }^{\infty }{s}^2\left[k\right]$$
  if the sum is finite.
• Power of a discrete time signal $s\left[k\right]$ is
  $$P\left(s\right)=\underset{N\to \infty }{lim}\frac{1}{2N}\sum _{k=-N}^N{s}^2\left[k\right]$$
  if the limit exists.
• Power Spectral Density
• With input and output transforms $X\left(f\right)$ and $Y\left(f\right)$ of a linear filter with impulse response transform $H\left(f\right)$ (such that $Y\left(f\right)=H\left(f\right)X\left(f\right)$ ),
  $${\mathcal{P}}_y\left(f\right)={\mathcal{P}}_x\left(f\right){\left|H\left(f\right)\right|}^2,$$
  where the power spectral density (PSD) is defined as
  $${\mathcal{P}}_x\left(f\right)=\underset{T\to \infty }{lim}\frac{|X_T{\left(f\right)|}^2}{T}\left(\mathrm{Watts}/\mathrm{Hz}\right),$$
  where $\mathcal{F}\left\{x_T\left(t\right)\right\}=X_T\left(f\right)$ and
  $$x_T\left(t\right)=x\left(t\right)\Pi \left(\frac{t}{T}\right),$$
  where $\Pi (·)$ is the rectangular pulse [link] .

Z-transforms and properties

• Definition of the Z-transform
  $$X\left(z\right)=\mathcal{Z}\left\{x\left[k\right]\right\}=\sum _{k=-\infty }^{\infty }x\left[k\right]z^{-k}$$
• Time-shift property
• With $\mathcal{Z}\left\{x\right[k\left]\right\}=X\left(z\right)$ ,
  $$\mathcal{Z}\left\{x[k-\Delta ]\right\}=z^{-\Delta }X\left(z\right).$$
• Linearity property
• With $\mathcal{Z}\left\{x_i\left[k\right]\right\}=X_i\left(z\right)$ ,
  $$\mathcal{Z}\{a{x}_1\left[k\right]+b{x}_2\left[k\right]\}=a{X}_1\left(z\right)+b{X}_2\left(z\right).$$
• Final Value Theorem for $z$ -transforms If $X\left(z\right)$ converges for $\left|z\right|>1$ and all poles of $(z-1)X\left(z\right)$ are inside the unit circle, then
  $$\underset{k\to \infty }{lim}x\left[k\right]=\underset{z\to 1}{lim}(z-1)X\left(z\right).$$

Integral and derivative formulas

• Sifting property of impulse
  $${\int }_{-\infty }^{\infty }w\left(t\right)\delta (t-t_0)dt=w\left(t_0\right)$$
• Schwarz's inequality
  $${\left|{\int }_{-\infty }^{\infty },a,\left(x\right),b,\left(x\right),d,x\right|}^2\le \left\{{\int }_{-\infty }^{\infty },{\left|a\left(x\right)\right|}^2,d,x\right\}\left\{{\int }_{-\infty }^{\infty },{\left|b\left(x\right)\right|}^2,d,x\right\}$$
  and equality occurs only when $a\left(x\right)=k{b}^{*}\left(x\right)$ , where superscript $*$ indicates complex conjugation (i.e.,  ${(a+jb)}^{*}=a-jb$ ).
• Leibniz's rule
  $$\begin{array}{cc}\hfill \frac{{\displaystyle d\left[{\int }_{a\left(x\right)}^{b\left(x\right)},f,(\lambda ,x),d,\lambda \right]}}{{\displaystyle dx}}& =f(b\left(x\right),x)\frac{db\left(x\right)}{dx}-f(a\left(x\right),x)\frac{da\left(x\right)}{dx}\hfill \\ & +{\int }_{a\left(x\right)}^{b\left(x\right)}\frac{\partial f(\lambda ,x)}{\partial x}d\lambda \hfill \end{array}$$
• Chain rule of differentiation
  $$\frac{dw}{dx}=\frac{dw}{dy}\frac{dy}{dx}$$
• Derivative of a product
  $$\frac{d}{dx}\left(wy\right)=w\frac{dy}{dx}+y\frac{dw}{dx}$$
• Derivative of signal raised to a power
  $$\frac{d}{dx}\left(y^n\right)=n{y}^{n-1}\frac{dy}{dx}$$
• Derivative of cosine
  $$\frac{d}{dx}\left(\cos ,\left(,y,\right)\right)=-\left(\sin \left(y\right)\right)\frac{dy}{dx}$$
• Derivative of sine
  $$\frac{d}{dx}\left(\sin ,\left(,y,\right)\right)=\left(\cos \left(y\right)\right)\frac{dy}{dx}$$

Matrix algebra

• Transpose transposed
  $${\left(A^T\right)}^T=A$$
• Transpose of a product
  $${\left(AB\right)}^T=B^T{A}^T$$
• Transpose and inverse commutativity If $A^{-1}$ exists,
  $${\left(A^T\right)}^{-1}={\left(A^{-1}\right)}^T.$$
• Inverse identity If $A^{-1}$ exists,
  $$A^{-1}A=A{A}^{-1}=I.$$