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S. Mandayam/ ECOMMS/ECE Dept./Rowan University Electrical Communications Systems ECE.09.331 Spring 2008 Shreekanth Mandayam ECE Department Rowan University http://engineering.rowan.edu/~shreek/spring08/ecomms/ Lecture 5a February 19, 2008

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S. Mandayam/ ECOMMS/ECE Dept./Rowan UniversityPlan Sampling Recall: Nyquist Sampling Impulse Sampling Dimensionality Theorem Discrete Fourier Transform (DFT) Fast Fourier Transform (FFT) Relation between CFT and DFT Baseband and Bandpass Signals Modulation Battle Plan for Analyzing Comm Systems

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S. Mandayam/ ECOMMS/ECE Dept./Rowan University ECOMMS: Topics

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S. Mandayam/ ECOMMS/ECE Dept./Rowan University Nyquist Sampling Theorem Any physical waveform can be represented by where If w ( t ) is band-limited to B Hz and

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S. Mandayam/ ECOMMS/ECE Dept./Rowan University What does this mean? 1/f s 2/f s 3/f s 4/f s 5/f s w(t) t a 3 = w(3/f s ) If then we can reconstruct w(t) without error by summing weighted, delayed sinc pulses weight = w(n/f s ) delay = n/f s We need to store only “samples” of w(t), i.e., w(n/f s ) The sinc pulses can be generated as needed (How?) Matlab Demo: sampling.m

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S. Mandayam/ ECOMMS/ECE Dept./Rowan University Impulse Sampling How do we mathematically represent a sampled waveform in the Time Domain? Frequency Domain?

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S. Mandayam/ ECOMMS/ECE Dept./Rowan University Sampling: Spectral Effect w(t) t w s (t ) t f -B 0 B |W(f)| f |W s (f) | -2f s -f s 0 f s 2 f s (-f s -B) -(f s +B) -B B (f s -B) (f s +B) F F Original Sampled

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S. Mandayam/ ECOMMS/ECE Dept./Rowan University Spectral Effect of Sampling Spectrum of a “sampled” waveform Spectrum of the “original” waveform replicated every f s Hz =

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S. Mandayam/ ECOMMS/ECE Dept./Rowan UniversityAliasing If f s < 2B, the waveform is “undersampled” “aliasing” or “spectral folding” How can we avoid aliasing? Increase f s “Pre-filter” the signal so that it is bandlimited to 2B < f s

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S. Mandayam/ ECOMMS/ECE Dept./Rowan University Dimensionality Theorem A real waveform can be completely specified by N = 2BT 0 independent pieces of information over a time interval T 0 N: Dimension of the waveform B: Bandwidth BT 0 : Time-Bandwidth Product Memory calculation for storing the waveform f s >= 2B At least N numbers must be stored over the time interval T0 = n/f s

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S. Mandayam/ ECOMMS/ECE Dept./Rowan University 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 time intervals Equal frequency intervals n = 0, 1, 2,….., N-1 k = 0, 1, 2,….., N-1

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S. Mandayam/ ECOMMS/ECE Dept./Rowan University Importance of the DFT Allows time domain / spectral domain transformations using discrete arithmetic operations Computational Complexity Raw DFT: N 2 complex operations (= 2N 2 real operations) Fast Fourier Transform (FFT): N log 2 N real operations Fast Fourier Transform (FFT) Cooley and Tukey (1965), ‘Butterfly Algorithm”, exploits the periodicity and symmetry of e -j2 kn/N VLSI implementations: FFT chips Modern DSP

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S. Mandayam/ ECOMMS/ECE Dept./Rowan University 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) n=0 1 2 3 4 n=N f=0 f = f s Need to know f s

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S. Mandayam/ ECOMMS/ECE Dept./Rowan University DFT Properties DFT is periodic X[n] = X[n+N] = X[n+2N] = ……… I-DFT is also periodic! x[k] = x[k+N] = x[k+2N] = ………. Where are the “low” and “high” frequencies on the DFT spectrum? n=0 N/2 n=N f=0 f s /2 f = f s

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S. Mandayam/ ECOMMS/ECE Dept./Rowan University Relation between CFT and DFT Windowing Sampling Generation of Periodic Samples

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S. Mandayam/ ECOMMS/ECE Dept./Rowan University Baseband and Bandpass Signals Baseband signals: spectral magnitude is non-zero only near the origin and is zero (or negligible) elsewhere Bandpass signals: spectral magnitude is non-zero only near the vicinity of f = ± f c, were f c >> 0 - f m 0 +f m W(f) f - f c 0 +f c W(f) f AF Signals RF Signals Carrier Frequency

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S. Mandayam/ ECOMMS/ECE Dept./Rowan UniversityModulation Modulating Signal Message Signal m(t) (Baseband) Modulated Signal s(t) (Bandpass) Information Modulation Frequency Translation What is it? How is it done? Modulator message: m(t) carrier: c(t) s(t): radio signal

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S. Mandayam/ ECOMMS/ECE Dept./Rowan University Why Modulate? Antenna size considerations Narrow banding Frequency multiplexing Common processing

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S. Mandayam/ ECOMMS/ECE Dept./Rowan University Battle Plan for Analyzing any Comm. Sys. Signals Systems Complex Envelope Time Domain Spectrum Power Performance Transmitters Receivers Standards Modulation Index Efficiency Bandwidth Noise

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S. Mandayam/ ECOMMS/ECE Dept./Rowan University Amplitude Modulation (AM) Signals Systems Complex Envelope Time Domain Spectrum Power Performance Transmitters Receivers Standards Modulation Index Efficiency Bandwidth Noise

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S. Mandayam/ ECOMMS/ECE Dept./Rowan University AM Spectrum -B 0 +B |M(f)| f -f c -B -f c -f c +B 0 f c -B f c f c +B |S(f)| f Message Spectrum AM Spectrum

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S. Mandayam/ ECOMMS/ECE Dept./Rowan University Single Tone AM s(t) = A c [1 + A m cos(2 f m t)]cos(2 f c t) c(t) = A c cos(2 f c t) m(t) = A m cos(2 f m t) t m(t) AmAm t c(t) AcAc s(t) t AcAc A c [1+A m ]

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S. Mandayam/ ECOMMS/ECE Dept./Rowan University Spectrum of Single Tone AM Matlab Demo Instrument Demo http://engineering.rowan.edu/~shreek/spring08/ecomms/demos/am.m

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S. Mandayam/ ECOMMS/ECE Dept./Rowan University Percentage Modulation s(t) t AcAc A max = A c [1+A m ] A min = A c [1-A m ]

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S. Mandayam/ ECOMMS/ECE Dept./Rowan UniversitySummary

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