Electrical Communication Systems ECE.09.433 Spring 2019 Lecture 2a January 29, 2019 Shreekanth Mandayam ECE Department Rowan University http://engineering.rowan.edu/~shreek/spring19/ecomms/

Plan Digital and Analog Communications Systems Properties of Signals and Noise Terminology Power and Energy Signals Recall: Continuous Fourier Transform (CFT) Discrete Fourier Transform (DFT) Recall: CFT’s (spectra) of common waveforms Impulse Sinusoid Rectangular Pulse

ECOMMS: Topics

Communications Systems 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

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

Power and Energy Signals 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 Although “real” signals are energy signals, we analyze them pretending they are power signals!

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

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

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

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

Summary