DIGITAL SIGNAL PROCESSING CCC-INAOE AUTUMN 2015

Transcription

1 DIGITAL SIGNAL PROCESSING CCC-INAOE AUTUMN 2015 Fourier Transform Properties Claudia Feregrino-Uribe & Alicia Morales Reyes Original material: Rene Cumplido "The Scientist and Engineer's Guide to Digital Signal Processing, copyright by Steven W. Smith."

2 Fourier Transform Pairs For every time domain waveform there is a corresponding frequency domain waveform, and vice versa. Waveforms that correspond to each other in this manner are called Fourier transform pairs

3 Compression and expansion If an event happens faster in time, It must be composed of high frequencies If an event happens slow in time, it must be composed by low frequencies Extremes: If a time domain signal is compressed to become an impulse, its frequency spectrum is expanded it becomes a constant value If a time domain signal is expanded to become a constant value, its frequency spectrum is compressed to become an impulse

4 Compression and expansion

5 Compression and expansion Compression in time domain correspond to an expansion in frequency domain

6 Compression and expansion Expansion in time domain correspond to compression in frequency domain

7 Fourier Transform Pairs For every time domain waveform there is a corresponding frequency domain waveform, and vice versa. Waveforms that correspond to each other in this manner are called Fourier transform pairs

8 Delta Function Pairs Discrete simple waveform Equally simple Fourier transform pair

9 Delta Function Pairs Delta function is shifted 4 samples to the right Magnitude is not affected Phase is changed by a linear component Negative frequencies are redundant information

10 Delta Function Pairs Delta function is shifted 8 samples to the right Magnitude is not affected Phase is changed by a linear component Negative frequencies are redundant information

11 Delta Function Pairs Delta function in rectangular representation Each sample in time domain results in a cosine wave (real part) and a negative sine wave (imaginary part) in frequency domain Duality: Each sample in DFT s frequency domain corresponds to a sinusoid in the time domain and viceversa

12 Delta Function Pairs

13 Delta Function Pairs Each sample in time domain results in a cosine wave and a negative sine wave added to the real part in frequency domain Sinusoids frequency is provided by corresponding sample number Sinusoids amplitude is given by time domain sample

14 The Sinc function Transform pair: Rectangular pulse Sinc function: sin(x)/x Sinc function is a sine wave that decays in amplitude as 1/x

15 The Sinc function Phase shift of pi for negative Magnitude in unwrapped A single pulse in frequency domain because of time periodicity Unwrapped means allowing positive and negative values

16 The Sinc function Aliasing occurs in discrete signals M, number of samples In rectangular pulse Rectangular pulse is shifted Magnitude is not changed Phase is changed by a linear component Magnitude is determined by equation Phase correspond to shift in time domain Radians

17 The Sinc function Discrete domain Aliasing occurs in discrete signals N/2 + 1 samples in frequency Continuous domain

18 The Sinc function Removing aliasing Pi*k/N Pi*f Zero division x becomes very small y(x)=sin(x) approaches y(x)=x

19 The Sync function Adding sinusoid samples within rectangular pulse Zero crossings Rectangular pulse: 20 samples wide 1 st crossing in frequency domain at frequency of 1 complete cycle in 20 samples 2 nd crossing in frequency domain at frequency of 2 complete cycles in 20 samples

20 Other transform pairs Duality Sinc function is the filter kernel for the perfect low-pass filter

21 Other transform pairs 2M-1 point triangle in time domain is formed by convolving two M point rectangular pulses Convolution in time domain > multiplication in frequency domain Squared sinc function

22 Other transform pairs Ignoring aliasing, Gaussian in time domain is a Gaussian in frequency domain

23 Other transform pairs

24 Gibbs effect

25 Gibbs effect

26 Harmonics

27 Harmonics

28 Harmonics

29 Harmonics Aliasing induced by harmonics

30 Next Fast Fourier Transform