Electrical Communications Systems ECE.09.433

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

2. Plan • Digital and Analog Communications Systems • Properties of Signals and Noise • Terminology • Power and Energy Signals • Continuous Fourier Transform (CFT)

3. ECOMMS: Topics

4. Digital Finite set of messages (signals) inexpensive/expensive privacy & security data fusion error detection and correction More bandwidth More overhead (hw/sw) Analog Continuous set of messages (signals) Legacy Predominant Inexpensive Communications Systems

5. Signal Properties: Terminology • Waveform • Time-average operator • Periodicity • DC value • Power • RMS Value • Normalized Power • Normalized Energy

6. Power Signal Infinite duration Normalized power is finite and non-zero Normalized energy averaged over infinite time is infinite Mathematically tractable Energy Signal Finite duration Normalized energy is finite and non-zero Normalized power averaged over infinite time is zero Physically realizable Power and Energy Signals • Although “real” signals are energy signals, we analyze them pretending they are power signals!

7. The Decibel (dB) • Measure of power transfer • 1 dB = 10 log10 (Pout / Pin) • 1 dBm = 10 log10 (P / 10-3) where P is in Watts • 1 dBmV = 20 log10 (V / 10-3) where V is in Volts

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

9. 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. 52 FT Theorems

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

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

12. 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

13. 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

14. 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

15. Summary