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 1997-1998 by Steven W. Smith."
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
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
Compression and expansion
Compression and expansion Compression in time domain correspond to an expansion in frequency domain
Compression and expansion Expansion in time domain correspond to compression in frequency domain
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
Delta Function Pairs Discrete simple waveform Equally simple Fourier transform pair
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
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
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
Delta Function Pairs
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
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
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
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
The Sinc function Discrete domain Aliasing occurs in discrete signals N/2 + 1 samples in frequency Continuous domain
The Sinc function Removing aliasing Pi*k/N Pi*f Zero division x becomes very small y(x)=sin(x) approaches y(x)=x
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
Other transform pairs Duality Sinc function is the filter kernel for the perfect low-pass filter
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
Other transform pairs Ignoring aliasing, Gaussian in time domain is a Gaussian in frequency domain
Other transform pairs
Gibbs effect
Gibbs effect
Harmonics
Harmonics
Harmonics
Harmonics Aliasing induced by harmonics
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