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Electrical Communications Systems ECE.09.433

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 Impulse Sinusoid Rectangular Pulse Discrete Fourier Transform How to get the frequency axis in the DFT.

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Electrical Communications Systems ECE.09.433

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

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

  3. ECOMMS: Topics

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

  8. 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

  9. 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

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

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

  12. Summary

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