Fundamentals of Electrical Communications Systems: Fourier Transform & Sampling

1. Electrical Communications SystemsECE.09.331Spring 2008 Lecture 3bFebruary 6, 2008 Shreekanth Mandayam ECE Department Rowan University http://engineering.rowan.edu/~shreek/spring08/ecomms/

2. Plan • Recall: 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)

3. ECOMMS: Topics

4. Continuous Fourier Transform (CFT) Frequency, [Hz] Phase Spectrum Amplitude Spectrum Inverse Fourier Transform (IFT) Definitions See p. 45 Dirichlet Conditions

5. CFT’s of Common Waveforms • Impulse (Dirac Delta) • Sinusoid • Rectangular Pulse Matlab Demo: recpulse.m

6. FS: Periodic Signals CFT: Aperiodic Signals CFT for Periodic Signals Recall: • We want to get the CFT for a periodic signal • What is ?

7. Sine Wave w(t) = A sin (2pf0t) Square Wave A -A T0/2 T0 CFT for Periodic Signals Instrument Demo

8. 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 Sampling • Can a waveform be both time-limited and band-limited?

9. Nyquist Sampling Theorem • Any physical waveform can be represented by • where • If w(t) is band-limited to B Hz and

10. 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

11. Impulse Sampling • How do we mathematically represent a sampled waveform in the • Time Domain? • Frequency Domain?

12. |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

13. Spectrum of a “sampled” waveform Spectrum of the “original” waveform replicated every fs Hz = Spectral Effect of Sampling

14. 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

15. 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

16. 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

17. 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

18. 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

19. Summary