Useful Transforms and Relationships - DePiero, CalPoly State University DTFT: Discrete-Time Fourier Transform

][)( nhFH where +== = nn FnjenhFH 2][)( Principle Range: 5.05.0 F , F is a continuous variable for digital frequency )(FH is the frequency response of a system with impulse response h(n) If x(n) = A cos(2 pi F0 n) then y(n) = A )( 0FH cos(2 pi F0 n + )( 0FH ) Note that )0(H is a sum, not a mean (there is a scaling by N with discrete FTs) Not computationally feasible

DFT: Discrete Fourier Transform

][)( nhkHN where == = 10 /2][)( Nnn NknjenhkH for 10 Nk

k is a sample index for digital frequency. N samples present in time and frequency domains. H(k) consists of samples of H(F): NkFforFHkH kk /)()( == Computationally feasible

FFT: Fast Fourier Transform

Described by a butterfly diagram, no formula. Variables Nk, as with DFT. Also NkjkN eW /2= In FFT rN 2= , so signal may need to be zero padded. Computationally efficient!

Z-Transform

][)( nhzHz with +== = nn nznhzH 0 ][)(

FpijezforFHzH 2)()( == Relationship Between )(zH and Difference Equation for a 2nd Order System

==== = 1110 ][][][ Nkk kMkk k knyAknxBny , M=N=3 for 2nd Order

)1()1(

)1()1(

1)( 1

2

12

11

11

22

11

22

110

=++

++=zz

zz

zAzAzBzBBzH

, are zeros and are poles Relationship between Frequency Axes and Unit Circle in Z Plane (S = sample rate in Hz)

0 S/2 S -S/2

f (Hz)

0 0.1 - 5.0

F (cycles/sample)

5.0

0,1 == Fz25.0, == Fjz

5.0,1 == Fzk

0 N/2 N

-1

W14 -1

-1

-1

x[0]

x[2]

x[1]

x[3]

X(0)

X(1)

X(2)

X(3)

x[n] X(k) FFT Butterfly, N=4