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m N−1 k=0 −2πjmk/N k (N/2) −1 k=0 −2πjkm/ (N/2) 2k −2πjm/N (N/2) −1 k=0 −2πjkm/ (N/2) 2k+1

1. This document provides a formula sheet for digital signal processing. It includes formulas for the discrete Fourier transform (DFT), inverse DFT, discrete cosine transform (DCT), z-transform, frequency response, and other common digital signal processing transforms and operations. 2. It also lists some common MATLAB commands that may be useful for digital signal processing, including commands for windows, filters, fast Fourier transforms, and other transforms. 3. The document contains two tables - one with a brief overview of common z-transforms and their regions of convergence, and another with examples of Fourier transforms of some common signals.

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Lulzim Lumi
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39 views2 pages

m N−1 k=0 −2πjmk/N k (N/2) −1 k=0 −2πjkm/ (N/2) 2k −2πjm/N (N/2) −1 k=0 −2πjkm/ (N/2) 2k+1

1. This document provides a formula sheet for digital signal processing. It includes formulas for the discrete Fourier transform (DFT), inverse DFT, discrete cosine transform (DCT), z-transform, frequency response, and other common digital signal processing transforms and operations. 2. It also lists some common MATLAB commands that may be useful for digital signal processing, including commands for windows, filters, fast Fourier transforms, and other transforms. 3. The document contains two tables - one with a brief overview of common z-transforms and their regions of convergence, and another with examples of Fourier transforms of some common signals.

Uploaded by

Lulzim Lumi
Copyright
© Attribution Non-Commercial (BY-NC)
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Formula Sheet

Digital Signal Processing


1. Ym =

PN 1 2jmk/N
e
y
k=0

P(N/2)1 2jkm/(N/2)
e
y

2. Ym = k=0
N even.
3. yk =

2k

PN 1 +2jkm/N
e
Y

1
N

m=0

+ e2jm/N

P(N/2)1 2jkm/(N/2)
e
y
k=0

4. VN (x) = cos(N cos1 (x))


5. cos(2) = 2 cos2 () 1
6. z =

2+T s
2T s

7. s =

2 z1
T z+1

8. analog =

2
T

9. |H()|2 =
10. |H()|2 =
11. T (z) =

tan(fdigital T )
1
1+(/c )2N
1
1+2 VN2 (/c )

zz0
1zz0

12. Y(f) = F(y(t))(f )


13. y(t) =

e2jf t y(t) dt

e2jf t Y (f ) df

14. Phase Delay = H (Fc )/(2Fc )


15. Group Delay = 0H (Fc )/(2)
16. Here are some MATLAB commands you may find helpful:

window
butter
fft
cheby1
cheby2
psd
hann
bartlett
ifft
1

2k+1 ,

Series
uk = {. . . , 0, 1, 1, . . .}
ak uk
{kak }

Z-Transform
z
z1
z
za
d
z dz
A(z)

ROC
|z| > 1
|z| > |a|
essentially unchanged

Table 1: A Very Brief Table of Z-Transforms.

Function
cos(2Fc t)
sin(2F
c t)
(
1 |t| B/2
y(t) =
0 |t| > B/2

Fourier Transform
((f Fc ) + (f + Fc ))/2
((f Fc ) (f + Fc ))/(2j)
Y (f ) =

sin(Bf )
f

y(t) =
P

sin(Bt)
t

1 |f | B/2
0 |f | > B/2
P
Fs (f kFs )
Y 0 (f )/(2j)

Y (f ) =

(t kTs )
ty(t)

k=

Table 2: A Very Brief Table of Fourier Transforms.

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