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This lecture covers topics such as time-limited and band-limited waveforms, Nyquist sampling theorem, impulse sampling, discrete Fourier transform (DFT), and fast Fourier transform (FFT). MATLAB demos provide practical examples.
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Electrical Communications SystemsECE.09.331Spring 2011 Lecture 3bFebruary 2, 2011 Shreekanth Mandayam ECE Department Rowan University http://engineering.rowan.edu/~shreek/spring11/ecomms/
Plan • Recall: Sampling • Time-limited and Band-limited waveforms • Nyquist Sampling • Impulse Sampling • Dimensionality Theorem • Discrete Fourier Transform (DFT) • Fast Fourier Transform (FFT) • Relation between CFT and DFT/FFT
Time-limited waveform w(t) = 0; |t| > T Band-limited waveform W(f)=F{(w(t)}=0; |f| > B W(f) w(t) -B B f -T T t Recall: Sampling • Can a waveform be both time-limited and band-limited?
Nyquist Sampling Theorem • Any physical waveform can be represented by • where • If w(t) is band-limited to B Hz and
a3 = w(3/fs) w(t) t 1/fs 2/fs 3/fs 4/fs 5/fs What does this mean? • If then we can reconstruct w(t) without error by summing weighted, delayed sinc pulses • weight = w(n/fs) • delay = n/fs • We need to store only “samples” of w(t), i.e., w(n/fs) • The sinc pulses can be generated as needed (How?) Matlab Demo: sampling.m
Impulse Sampling • How do we mathematically represent a sampled waveform in the • Time Domain? • Frequency Domain?
|W(f)| F F w(t) -B 0 B t f |Ws(f)| ws(t) -2fs -fs 0 fs 2 fs t f (-fs-B) -(fs +B) -B B (fs -B) (fs +B) Sampling: Spectral Effect Original Sampled
Spectrum of a “sampled” waveform Spectrum of the “original” waveform replicated every fs Hz = Spectral Effect of Sampling
Aliasing • If fs < 2B, the waveform is “undersampled” • “aliasing” or “spectral folding” • How can we avoid aliasing? • Increase fs • “Pre-filter” the signal so that it is bandlimited to 2B < fs
Dimensionality Theorem • A real waveform can be completely specified by N = 2BT0 independent pieces of information over a time interval T0 • N: Dimension of the waveform • B: Bandwidth • BT0: Time-Bandwidth Product • Memory calculation for storing the waveform • fs >= 2B • At least N numbers must be stored over the time interval T0 = n/fs
Equal time intervals Discrete Fourier Transform (DFT) • Discrete Domains • Discrete Time: k = 0, 1, 2, 3, …………, N-1 • Discrete Frequency: n = 0, 1, 2, 3, …………, N-1 • Discrete Fourier Transform • Inverse DFT Equal frequency intervals n = 0, 1, 2,….., N-1 k = 0, 1, 2,….., N-1
Importance of the DFT • Allows time domain / spectral domain transformations using discrete arithmetic operations • Computational Complexity • Raw DFT: N2 complex operations (= 2N2 real operations) • Fast Fourier Transform (FFT): N log2 N real operations • Fast Fourier Transform (FFT) • Cooley and Tukey (1965), ‘Butterfly Algorithm”, exploits the periodicity and symmetry of e-j2pkn/N • VLSI implementations: FFT chips • Modern DSP
n=0 1 2 3 4 n=N f=0 f = fs How to get the frequency axis in the DFT • The DFT operation just converts one set of number, x[k] into another set of numbers X[n] - there is no explicit definition of time or frequency • How can we relate the DFT to the CFT and obtain spectral amplitudes for discrete frequencies? (N-point FFT) Need to know fs
n=0 N/2 n=N f=0 fs/2 f = fs DFT Properties • DFT is periodic X[n] = X[n+N] = X[n+2N] = ……… • I-DFT is also periodic! x[k] = x[k+N] = x[k+2N] = ………. • Where are the “low” and “high” frequencies on the DFT spectrum? Matlab Demo: dft.m
Relation between CFT and DFT • Windowing • Sampling • Generation of Periodic Samples Matlab Demo: myspectrum.m
