Digital Signal Processing

Discrete Fourier TransformDiscrete Fourier Transform

NNekxN

kxDFnx

NNenxkX

N

k

Nknj

NknjN

n

,,2,1,0,][1

][

,,2,1,0,

0

/21

/21

0

Discrete Fourier Transform

Inverse Discrete Fourier Transform

Properties of DFTProperties of DFT

• DFT has the same number of datapoints as the signal

• The signal is assumed to be periodic with a period of N

• X[k] corresponds to the amplitude of the signal at frequency f=k/(NT)

• The frequency resolution of the DFT is f=1/(NT), i.e. the # of samples determines the frequency resolution

Steps for Calculating DFTSteps for Calculating DFT

• Determine the resolution required for the DFT, establish a lower limit on the # of samples required, N.

• Determine the sampling frequency to avoid aliasing

• Accumulate N samples

• Calculate DFT

Matlab Example of FFTMatlab Example of FFT

Digital FilteringDigital Filtering

a1*y(n) = b1*x(n) +b2*x(n-1) + ... + bnb+1x(n-nb) - a2*y(n-1) - ... – ana+1*y(n-na)

A=[a1, a2, ..., ana+1]

B=[b1, b2, ..., bnb+1]

X=[x(n-nb), ..., x(n-1), x(n)]: input signal

Filter parameters

Y=[y(n-na), ..., y(n-1), y(n)]: filtered signal

Ideal FiltersIdeal Filters

• Low pass filter

• High pass filter

• Bandpass filter

• Bandstop filter

Common FiltersCommon Filters

• Butterworth filter: N

c

H2

1

1

• Chebyshev filter:

cn

cn

nC

C

H

1

22

coscos,

1

1

Comparison of Common FiltersComparison of Common Filters

MATLAB example of FilteringMATLAB example of Filtering

MATLAB Example of UndersamplingMATLAB Example of Undersampling