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S. Mandayam/ ECOMMS/ECE Dept./Rowan University Electrical Communications Systems 0909.331.01 Spring 2005 Shreekanth Mandayam ECE Department Rowan University.

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Presentation on theme: "S. Mandayam/ ECOMMS/ECE Dept./Rowan University Electrical Communications Systems 0909.331.01 Spring 2005 Shreekanth Mandayam ECE Department Rowan University."— Presentation transcript:

1 S. Mandayam/ ECOMMS/ECE Dept./Rowan University Electrical Communications Systems 0909.331.01 Spring 2005 Shreekanth Mandayam ECE Department Rowan University http://engineering.rowan.edu/~shreek/spring05/ecomms/ Lecture 3b February 16, 2005

2 S. Mandayam/ ECOMMS/ECE Dept./Rowan UniversityPlan 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

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

4 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

5 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 =

6 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

7 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

8 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

9 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

10 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

11 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

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

13 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

14 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

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

16 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

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


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