S. Mandayam/ ECOMMS/ECE Dept./Rowan University Electrical Communications Systems ECE Spring 2007 Shreekanth Mandayam ECE Department Rowan University Lecture 2b January 24, 2007

S. Mandayam/ ECOMMS/ECE Dept./Rowan UniversityPlan CFT’s (spectra) of common waveforms Impulse Sinusoid Rectangular Pulse 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)

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

S. Mandayam/ ECOMMS/ECE Dept./Rowan University Fourier Series Exponential Representation Periodic Waveform w(t) t |W(n)| f -3f 0 -2f 0 -f 0 f 0 2f 0 3f 0 2-Sided Amplitude Spectrum f 0 = 1/T 0 ; T 0 = period T0T0

S. Mandayam/ ECOMMS/ECE Dept./Rowan University Fourier Transform Fourier Series of periodic signals finite amplitudes spectral components separated by discrete frequency intervals of f 0 = 1/T 0 We want a spectral representation for aperiodic signals Model an aperiodic signal as a periodic signal with T > infinity Then, f > 0 The spectrum is continuous!

S. Mandayam/ ECOMMS/ECE Dept./Rowan University Continuous Fourier Transform We want a spectral representation for aperiodic signals Model an aperiodic signal as a periodic signal with T > infinity Then, f > 0 The spectrum is continuous! t T 0 Infinity w(t) Aperiodic Waveform |W(f)| f f 0 0

S. Mandayam/ ECOMMS/ECE Dept./Rowan UniversityDefinitions Continuous Fourier Transform (CFT) Frequency, [Hz] Amplitude Spectrum Phase Spectrum Inverse Fourier Transform (IFT) See p. 45 Dirichlet Conditions

S. Mandayam/ ECOMMS/ECE Dept./Rowan University Properties of FT’s If w(t) is real, then W(f) = W*(f) If W(f) is real, then w(t) is even If W(f) is imaginary, then w(t) is odd Linearity Time delay Scaling Duality See p. 50 FT Theorems

S. Mandayam/ ECOMMS/ECE Dept./Rowan University CFT’s of Common Waveforms Impulse (Dirac Delta) Sinusoid Rectangular Pulse Matlab Demo: recpulse.m

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 UniversitySummary