Digital Signal Processing/Transforms

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This page lists some of the transforms from the book, explains their uses, and lists some transform pairs of common functions.

Continuous-Time Fourier Transform (CTFT)[edit]


[CTFT]

\mathcal{F}(\omega) = \int f(t) e^{j \omega t} dt

CTFT Table[edit]

  Time Domain Frequency Domain
x(t) = \mathcal{F}^{-1}\left\{ X(\omega) \right\} X(\omega) = \mathcal{F} \left\{ x(t) \right\}
1  X(j \omega)=\int_{-\infty}^\infty x(t) e^{-j \omega t}d t  x(t)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} X(j \omega)e^{j \omega t}d \omega
2  1 \,  2\pi\delta(\omega) \,
3  -0.5 + u(t) \,  \frac{1}{j \omega} \,
4  \delta (t) \,  1 \,
5  \delta (t-c) \,  e^{-j \omega c} \,
6  u(t) \,  \pi \delta(\omega)+\frac{1}{j \omega} \,
7  e^{-bt}u(t) \, (b > 0)  \frac{1}{j \omega+b} \,
8  \cos \omega_0 t \,  \pi \left[ \delta(\omega+\omega_0)+\delta(\omega-\omega_0) \right] \,
9  \cos (\omega_0 t + \theta) \,  \pi \left[ e^{-j \theta}\delta(\omega+\omega_0)+e^{j \theta}\delta(\omega-\omega_0) \right] \,
10  \sin \omega_0 t \,  j \pi \left[ \delta(\omega +\omega_0)-\delta(\omega-\omega_0) \right] \,
11  \sin (\omega_0 t + \theta) \,  j \pi \left[ e^{-j \theta}\delta(\omega +\omega_0)-e^{j \theta}\delta(\omega-\omega_0) \right] \,
12  \mbox{rect} \left( \frac{t}{\tau} \right) \,  \tau \mbox{sinc} \left( \frac{\tau \omega}{2 \pi} \right) \,
13  \tau \mbox{sinc} \left( \frac{\tau t}{2 \pi} \right) \,  2 \pi \mbox{rect} \left( \frac{ \omega }{ \tau } \right) \,
14  \left( 1-\frac{2 |t|}{\tau} \right) \mbox{rect} \left( \frac{ t }{ \tau } \right) \,  \frac{\tau}{2} \mbox{sinc}^2 \left( \frac{\tau \omega}{4 \pi} \right) \,
15  \frac{\tau}{2} \mbox{sinc}^2 \left( \frac{\tau t}{4 \pi} \right) \,  2 \pi \left( 1-\frac{2|\omega|}{\tau} \right) \mbox{rect} \left( \frac{ \omega }{ \tau } \right) \,
16  e^{-a|t|}, \Re\{a\}>0 \,  \frac{2a}{a^2 + \omega^2} \,
Notes:
  1.  \mbox{sinc}(x)=\sin(x)/x
  2.  \mbox{rect} \left( \frac{ t }{ \tau } \right) is the rectangular pulse function of width  \tau
  3.  u(t) is the Heavyside step function
  4.  \delta (t) is the Dirac delta function

Discrete-Time Fourier Transform (DTFT)[edit]

DTFT Table[edit]

The information in the table below may be inaccurate - The DTFT transfers into the periodic frequency domain. The signals shown in the table are not periodic, and hence they are obviously wrong. The table seems to list ordinary Fourier transforms for angular frequency, which coincidentally also uses the greek letter ω, although with a different meaning.

Time domain
 x(n) \, where n\in\Z
Frequency domain
  X(\omega) \, where \omega\in [-\pi,\pi)
Remarks
\delta (n) \! 1 \! Here \delta (n) \! represents the delta function
which is 1 if n=0 and zero otherwise.
\delta (n - k) \! e^{-i k \omega} \! for some k \in \mathbb{Z}
u(n)=\begin{cases}0 &\text{for } n\\ 1 &\text{for } n\geq 0\end{cases} \frac{1}{1-e^{-i \omega}} \!
e^{-ian} \!    \delta (\omega + a)  \, For any real number  a \,. Here \delta(\omega) is a periodic Dirac delta function. That is
\textstyle \delta(\omega)=\delta(\omega+2\pi), \textstyle\int_{-\pi}^\pi\delta(\omega)\,d\omega=1 and informally \delta(\omega)=0 if \omega\neq 0
\cos (a n) \! \frac{1}{2} \Big( \delta (\omega - a) + \delta (\omega + a) \Big) a \in \mathbb{R}
\sin (a n) \! \frac{1}{2 i} \Big( \delta (\omega - a) - \delta ( \omega + a) \Big) a \in \mathbb{R}
 \mathrm{rect} \left( { ( n - M/2 ) \over M  } \right)  { \sin\Big( \omega (M+1) / 2 \Big) \over \sin( \omega / 2 ) } \,  e^{ -i \omega M / 2 } M \in \mathbb{Z}
\operatorname{sinc} (a + n) e^{i a \omega} \! a \in \mathbb{R}
W\cdot \operatorname{sinc}^2(W n)\, { 1 \over 2\pi W}\cdot \operatorname{tri} \left( { \omega \over 2\pi W } \right) W \in \mathbb{Z}
W\cdot \operatorname{sinc} \Big( W (n + a)\Big) \operatorname{rect} \left( { \omega \over 2\pi W } \right) \cdot e^{j a \omega} a, W \in \mathbb{R}

0 < W \le 1

\frac{W}{(n + a)} \Big[ \cos \big( \pi W (n+a)\big) - \operatorname{sinc} \big( W (n+a)\big) \Big] j \omega \cdot \operatorname{rect} \left( { \omega \over \pi W } \right) e^{j a \omega} a, W \in \mathbb{R}
\frac{1}{\pi n^2} \Big[(-1)^n - 1\Big] | \omega | \!
\frac{C (A + B)}{2 \pi} \cdot \operatorname{sinc} \left( \frac{A - B}{2\pi} n \right) \cdot \operatorname{sinc} \left( \frac{A + B}{2\pi} n \right) Trapezoid signal.svg A, B \in \mathbb{R}

C \in \mathbb{C}

DTFT Properties[edit]

Property Time domain
x[n] \!
Frequency domain
X(\omega) \!
Remarks
Linearity a x[n] + b y[n] \!  a X(e^{i \omega}) + b Y(e^{i \omega}) \!
Shift in time x[n - k] \! X(e^{i \omega}) e^{-i \omega k} \! integer k
Shift in frequency x[n] e^{i a n} \! X(e^{i (\omega-a)}) \! real number a
Time reversal x[- n] \! X(e^{-i \omega}) \!
Time conjugation x[n]^* \! X(e^{-i \omega})^* \!
Time reversal & conjugation x[-n]^* \! X(e^{i \omega})^* \!
Derivative in frequency \frac{n}{i} x[n] \! \frac{d X(e^{i \omega})}{d \omega} \!
Integral in frequency \frac{i}{n} x[n] \! \int_{-\pi}^{\omega} X(e^{i \vartheta}) d \vartheta \!
Convolve in time x[n] * y[n] \! X(e^{i \omega}) \cdot Y(e^{i \omega}) \!
Multiply in time x[n] \cdot y[n] \! \frac{1}{2 \pi} X(e^{i \omega}) * Y(e^{i \omega}) \!
Correlation \rho_{xy} [n] = x[-n]^* * y[n] \! R_{xy} (\omega) = X(e^{i \omega})^* \cdot Y(e^{i \omega}) \!

Where:

  • *\! is the convolution between two signals
  • x[n]^*\! is the complex conjugate of the function x[n]
  • \rho_{xy} [n]\! represents the correlation between x[n] and y[n].

Discrete Fourier Transform (DFT)[edit]

DFT Table[edit]

Time-Domain
x[n]
Frequency Domain
X[k]
Notes
x_n\equiv\frac{1}{N}\sum_{k=0}^{N-1}X_k \cdot e^{i 2 \pi kn/N} X_k\equiv\sum_{n=0}^{N-1}x_n \cdot e^{-i 2 \pi kn/N} DFT Definition
x_n \cdot e^{i 2 \pi kn/N} \, X_{n-k}\, Shift theorem
x_{n-k}\, X_k \cdot e^{-i 2 \pi kn/N}
x_n \in \mathbf{R} X_k=X_{N-k}^*\, Real DFT
a^n\, \frac{1-a^N}{1-a \cdot e^{-i 2 \pi k/N} }  
{N-1 \choose n}\, \left(1+e^{-i 2 \pi k/N} \right)^{N-1}\,  

Z-Transform[edit]

Z-Transform Table[edit]

Here:

  • u[n]=1 for n>=0, u[n]=0 for n<0
  • \delta[n] = 1 for n=0, \delta[n] = 0 otherwise
Signal, x[n] Z-transform, X(z) ROC
1 \delta[n] \, 1\,  \mbox{all }z\,
2 \delta[n-n_0] \,  z^{-n_0} \,  z \neq 0\,
3 u[n] \,  \frac{1}{1-z^{-1} } |z| > 1\,
4 - u[-n-1] \,  \frac{1}{1 - z^{-1}} |z| < 1\,
5 n u[n] \,  \frac{z^{-1}}{( 1-z^{-1} )^2} |z| > 1\,
6  - n u[-n-1] \,  \frac{z^{-1} }{ (1 - z^{-1})^2 }  |z| < 1 \,
7 n^2 u[n] \,   \frac{ z^{-1} (1 + z^{-1} )}{(1 - z^{-1})^3} |z| > 1\,
8  - n^2 u[-n - 1] \,   \frac{ z^{-1} (1 + z^{-1} )}{(1 - z^{-1})^3} |z| < 1\,
9 n^3 u[n] \,  \frac{z^{-1} (1 + 4 z^{-1} + z^{-2} )}{(1-z^{-1})^4} |z| > 1\,
10 - n^3 u[-n -1] \,  \frac{z^{-1} (1 + 4 z^{-1} + z^{-2} )}{(1-z^{-1})^4} |z| < 1\,
11 a^n u[n] \,  \frac{1}{1-a z^{-1}}  |z| > |a|\,
12 -a^n u[-n-1] \,  \frac{1}{1-a z^{-1}} |z| < |a|\,
13 n a^n u[n] \,  \frac{az^{-1} }{ (1-a z^{-1})^2 } |z| > |a|\,
14 -n a^n u[-n-1] \,  \frac{az^{-1} }{ (1-a z^{-1})^2 }  |z| < |a|\,
15 n^2 a^n u[n] \,  \frac{a z^{-1} (1 + a z^{-1}) }{(1-a z^{-1})^3} |z| > |a|\,
16 - n^2 a^n u[-n -1] \,  \frac{a z^{-1} (1 + a z^{-1}) }{(1-a z^{-1})^3} |z| < |a|\,
17 \cos(\omega_0 n) u[n] \,  \frac{ 1-z^{-1} \cos(\omega_0) }{ 1-2z^{-1}\cos(\omega_0)+ z^{-2} }  |z| >1\,
18 \sin(\omega_0 n) u[n] \,  \frac{ z^{-1} \sin(\omega_0) }{ 1-2z^{-1}\cos(\omega_0)+ z^{-2} }  |z| >1\,
19 a^n \cos(\omega_0 n) u[n] \,  \frac{ 1-a z^{-1} \cos( \omega_0) }{ 1-2az^{-1}\cos(\omega_0)+ a^2 z^{-2} }  |z| > |a|\,
20 a^n \sin(\omega_0 n) u[n] \,  \frac{ az^{-1} \sin(\omega_0) }{ 1-2az^{-1}\cos(\omega_0)+ a^2 z^{-2} }  |z| > |a|\,

Bilinear Transform[edit]

see [1]

Discrete Cosine Transform (DCT)[edit]

Haar Transform[edit]