ECE 4371, Fall 2009 Zhu Han Department of Electrical and Computer Engineering Class 8 Sep. 17 th, 2007

Overview Midterm 1 for analog, up to class 7 (include class 7), 10/6 Homework: 2.24, 2.29, 2.32, 2.33, 2.46, 2.56, 2.59, due 10/ two lab reports before the midterm Sampling Theorem –Math –Reconstruction –Aliasing –Bandpass sampling theorem Pulse Amplitude Modulation Pulse Width Modulation Pulse Position Modulation Pulse Coded Modulation

Claude Elwood Shannon, Harry Nyquist

Pulse Modulation Introduction

2 Sampling Theorem

3 Sampling Theorem

4 Reconstruction Theorem

Interpolation If the sampling is at exactly the Nyquist rate, then

5 Sampling Theorem

Figure 3.3 (a) Spectrum of a signal. (b) Spectrum of an undersampled version of the signal exhibiting the aliasing phenomenon. 6 Aliasing

Figure 3.4 (a) Anti-alias filtered spectrum of an information-bearing signal. (b) Spectrum of instantaneously sampled version of the signal, assuming the use of a sampling rate greater than the Nyquist rate. (c) Magnitude response of reconstruction filter. 7 Anti-Alias Filter

Aliasing 2D example

Example: Aliasing of Sinusoidal Signals Frequency of signals = 500 Hz, Sampling frequency = 2000Hz

Example: Aliasing of Sinusoidal Signals Frequency of signals = 1100 Hz, Sampling frequency = 2000Hz

Example: Aliasing of Sinusoidal Signals Frequency of signals = 1500 Hz, Sampling frequency = 2000Hz

Example: Aliasing of Sinusoidal Signals Frequency of signals = 1800 Hz, Sampling frequency = 2000Hz

Example: Aliasing of Sinusoidal Signals Frequency of signals = 2200 Hz, Sampling frequency = 2000Hz

Bandpass Sampling (a) variable sample rate (b) maximum sample rate without aliasing (c) minimum sampling rate without aliasing

Bandpass Sampling A signal of bandwidth B, occupying the frequency range between f L and f L + B, can be uniquely reconstructed from the samples if sampled at a rate f S : f S >= 2 * (f2-f1)(1+M/N) where M=f 2 /(f 2 -f 1 ))-N and N = floor(f 2 /(f 2 -f 1 )), B= f 2 -f 1, f2=NB+MB.

Bandpass Sampling Theorem

Pulse-Amplitude Modulation 8

9 Pulse-Amplitude Modulation Frequency

Pulse Amplitude Modulation – Natural Sampling The circuit of Figure is used to illustrate pulse amplitude modulation (PAM). The FET is the switch used as a sampling gate. When the FET is on, the analog voltage is shorted to ground; when off, the FET is essentially open, so that the analog signal sample appears at the output. Op-amp 1 is a noninverting amplifier that isolates the analog input channel from the switching function. Op-amp 2 is a high input-impedance voltage follower capable of driving low- impedance loads (high “fanout”). The resistor R is used to limit the output current of op-amp 1 when the FET is “on” and provides a voltage division with r d of the FET. (r d, the drain-to-source resistance, is low but not zero)

The most common technique for sampling voice in PCM systems is to a sample-and- hold circuit. As seen in Figure, the instantaneous amplitude of the analog (voice) signal is held as a constant charge on a capacitor for the duration of the sampling period T s. This technique is useful for holding the sample constant while other processing is taking place, but it alters the frequency spectrum and introduces an error, called aperture error, resulting in an inability to recover exactly the original analog signal. The amount of error depends on how mach the analog changes during the holding time, called aperture time. To estimate the maximum voltage error possible, determine the maximum slope of the analog signal and multiply it by the aperture time DT in Figure Pulse Amplitude Modulation – Flat-Top Sampling

Recovering the original message signal m(t) from PAM signal 10

PAM, PWM, PPM, PCM

In pulse width modulation (PWM), the width of each pulse is made directly proportional to the amplitude of the information signal. In pulse position modulation, constant-width pulses are used, and the position or time of occurrence of each pulse from some reference time is made directly proportional to the amplitude of the information signal. PWM and PPM are compared and contrasted to PAM in Figure. Pulse Width and Pulse Position Modulation

Pulse Width Modulation

Pulse Position Modulation

Pulse Code Modulation (PCM) Pulse code modulation (PCM) is produced by analog-to-digital conversion process. Quantized PAM As in the case of other pulse modulation techniques, the rate at which samples are taken and encoded must conform to the Nyquist sampling rate. The sampling rate must be greater than, twice the highest frequency in the analog signal, f s > 2f A (max) Telegraph time-division multiplex (TDM) was conveyed as early as 1853, by the American inventor M.B. Farmer. The electrical engineer W.M. Miner, in PCM was invented by the British engineer Alec Reeves in 1937 in France.Alec Reeves1937 It was not until about the middle of 1943 that the Bell Labs people became aware of the use of PCM binary coding as already proposed by Alec Reeves.Bell Labs

Digital Modulation The input is discrete signal –Time sequences of pulses or symbols Offers many advantages –Robustness to channel impairments –Easier multiplexing of various sources of information: voice, data, video. –Can accommodate digital error-control codes –Enables encryption of the transferred signals u More secure link

Digital Modulation Example The modulating signal is represented as a time-sequence of symbols or pulses. Each symbol has m finite states: That means each symbol carries n bits of information where n = log 2 m bits/symbol T One symbol (has m states – voltage levels) (represents n = log 2 m bits of information) Modulator

Factors that Influence Choice of Digital Modulation Techniques A desired modulation scheme –Provides low bit-error rates at low SNRs u Power efficiency –Performs well in multipath and fading conditions –Occupies minimum RF channel bandwidth u Bandwidth efficiency –Is easy and cost-effective to implement Depending on the demands of a particular system or application, tradeoffs are made when selecting a digital modulation scheme.

Power Efficiency of Modulation Power efficiency is the ability of the modulation technique to preserve fidelity of the message at low power levels. Usually in order to obtain good fidelity, the signal power needs to be increased. –Tradeoff between fidelity and signal power –Power efficiency describes how efficient this tradeoff is made E b : signal energy per bit N 0 : noise power spectral density PER: probability of error

Bandwidth Efficiency of Modulation Ability of a modulation scheme to accommodate data within a limited bandwidth. Bandwidth efficiency reflect how efficiently the allocated bandwidth is utilized R: the data rate (bps) B: bandwidth occupied by the modulated RF signal

Shannon’s Bound There is a fundamental upper bound on achievable bandwidth efficiency. Shannon’s theorem gives the relationship between the channel bandwidth and the maximum data rate that can be transmitted over this channel considering also the noise present in the channel. Shannon’s Theorem C: channel capacity (maximum data-rate) (bps) B: RF bandwidth S/N: signal-to-noise ratio (no unit)