Electrical Communications Systems ECE.09.433 Signals and Spectra II Dr. Shreek Mandayam Electrical & Computer Engineering Rowan University

Plan CFT’s (spectra) of common waveforms Discrete Fourier Transform Impulse Sinusoid Rectangular Pulse Discrete Fourier Transform How to get the frequency axis in the DFT

ECOMMS: Topics

Continuous Fourier Transform Continuous Fourier Transform (CFT) Frequency, [Hz] Amplitude Spectrum Phase Inverse Fourier Transform (IFT) See p. 46 Dirichlet Conditions

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

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

Discrete Fourier Transform (DFT) Equal time intervals 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

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 1 2 3 4 n=N f=0 f = fs

All those signals………. Amplitude w(t) time Time Amplitude Signal 111 continuous continuous continuous-time analog signal discrete discrete discrete-time digital signal Cn 111 110 101 100 011 010 001 000 sampling Amplitude time discrete continuous discrete-time analog signal w(nTs) Ts sampling discrete continuous discrete-time sequence w[n] n=0 1 2 3 4 5 indexing

…..and all those transforms Sample in time, period = Ts Continuous-time analog signal w(t) Discrete-time analog sequence w [n] C D Continuous-variable Discrete-variable Laplace Transform W(s) Continuous Fourier Transform W(f) z-Transform W(z) Discrete-Time Fourier Transform W(W) Discrete Fourier Transform W(k) z = ejW s = jw w=2pf =2pf W = w Ts, scale amplitude by 1/Ts Sample in frequency, W = 2pn/N, N = Length of sequence

Summary