S. Mandayam/ ECOMMS/ECE Dept./Rowan University Electrical Communications Systems Spring 2005 Shreekanth Mandayam ECE Department Rowan University Lecture 3a February 15, 2005

S. Mandayam/ ECOMMS/ECE Dept./Rowan UniversityPlan CFT’s for periodic waveforms 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

S. Mandayam/ ECOMMS/ECE Dept./Rowan University ECOMMS: Topics

S. Mandayam/ ECOMMS/ECE Dept./Rowan University CFT for Periodic Signals Recall: CFT: Aperiodic Signals FS: Periodic Signals We want to get the CFT for a periodic signal What is ?

S. Mandayam/ ECOMMS/ECE Dept./Rowan University CFT for Periodic Signals Sine Wave w(t) = A sin (2  f 0 t) Square Wave A -A T 0 /2 T 0 Instrument Demo

S. Mandayam/ ECOMMS/ECE Dept./Rowan UniversitySampling Time-limited waveform w(t) = 0; |t| > T Band-limited waveform W(f)= F {(w(t)}=0; |f| > B -T T w(t) t -B B W(f) f Can a waveform be both time-limited and band-limited?

S. Mandayam/ ECOMMS/ECE Dept./Rowan University Nyquist Sampling Theorem Any physical waveform can be represented by where If w ( t ) is band-limited to B Hz and

S. Mandayam/ ECOMMS/ECE Dept./Rowan University What does this mean? 1/f s 2/f s 3/f s 4/f s 5/f s w(t) t a 3 = w(3/f s ) If then we can reconstruct w(t) without error by summing weighted, delayed sinc pulses weight = w(n/f s ) delay = n/f s We need to store only “samples” of w(t), i.e., w(n/f s ) The sinc pulses can be generated as needed (How?) Matlab Demo: sampling.m

S. Mandayam/ ECOMMS/ECE Dept./Rowan University Impulse Sampling How do we mathematically represent a sampled waveform in the Time Domain? Frequency Domain?

S. Mandayam/ ECOMMS/ECE Dept./Rowan University Sampling: Spectral Effect w(t) t w s (t ) t f -B 0 B |W(f)| f |W s (f) | -2f s -f s 0 f s 2 f s (-f s -B) -(f s +B) -B B (f s -B) (f s +B) F F Original Sampled

S. Mandayam/ ECOMMS/ECE Dept./Rowan University Spectral Effect of Sampling Spectrum of a “sampled” waveform Spectrum of the “original” waveform replicated every f s Hz =

S. Mandayam/ ECOMMS/ECE Dept./Rowan UniversityAliasing If f s < 2B, the waveform is “undersampled” “aliasing” or “spectral folding” How can we avoid aliasing? Increase f s “Pre-filter” the signal so that it is bandlimited to 2B < f s

S. Mandayam/ ECOMMS/ECE Dept./Rowan University Dimensionality Theorem A real waveform can be completely specified by N = 2BT 0 independent pieces of information over a time interval T 0 N: Dimension of the waveform B: Bandwidth BT 0 : Time-Bandwidth Product Memory calculation for storing the waveform f s >= 2B At least N numbers must be stored over the time interval T0 = n/f s

S. Mandayam/ ECOMMS/ECE Dept./Rowan University 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 time intervals Equal frequency intervals n = 0, 1, 2,….., N-1 k = 0, 1, 2,….., N-1

S. Mandayam/ ECOMMS/ECE Dept./Rowan University Importance of the DFT Allows time domain / spectral domain transformations using discrete arithmetic operations Computational Complexity Raw DFT: N 2 complex operations (= 2N 2 real operations) Fast Fourier Transform (FFT): N log 2 N real operations Fast Fourier Transform (FFT) Cooley and Tukey (1965), ‘Butterfly Algorithm”, exploits the periodicity and symmetry of e -j2  kn/N VLSI implementations: FFT chips Modern DSP

S. Mandayam/ ECOMMS/ECE Dept./Rowan University 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) n= n=N f=0 f = f s Need to know f s

S. Mandayam/ ECOMMS/ECE Dept./Rowan University 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? n=0 N/2 n=N f=0 f s /2 f = f s

S. Mandayam/ ECOMMS/ECE Dept./Rowan University Relation between CFT and DFT Windowing Sampling Generation of Periodic Samples

S. Mandayam/ ECOMMS/ECE Dept./Rowan UniversitySummary