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

ECE 4371, Fall 2009

Zhu Han

Department of Electrical and Computer Engineering

Class 8

Sep. 17th, 2007

OverviewOverview 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 10/1

4117 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 NyquistClaude Elwood Shannon, Harry Nyquist

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Pulse Modulation Introduction

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Figure 3.3 Figure 3.3 ((aa) Spectrum of a signal. () Spectrum of a signal. (bb) Spectrum of an undersampled version of the signal exhibiting ) Spectrum of an undersampled version of the signal exhibiting the aliasing phenomenon.the aliasing phenomenon.

6

Aliasing

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

7

Anti-Alias Filter

AliasingAliasing 2D example

Example: Aliasing of Sinusoidal SignalsExample: Aliasing of Sinusoidal Signals

Frequency of signals = 500 Hz, Sampling frequency = 2000Hz









Bandpass SamplingBandpass Sampling(a) variable sample rate

(b) maximum sample rate without aliasing(c) minimum sampling rate without aliasing

Bandpass SamplingBandpass Sampling A signal of bandwidth B, occupying the frequency range

between fL and fL + B, can be uniquely reconstructed from the samples if sampled at a rate fS :

fS >= 2 * (f2-f1)(1+M/N)

where M=f2/(f2-f1))-N and N = floor(f2/(f2-f1)),

B= f2-f1, f2=NB+MB.

Bandpass Sampling TheoremBandpass Sampling Theorem

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Pulse-Amplitude Modulation

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Pulse-Amplitude Modulation Frequency

Pulse Amplitude Modulation – Pulse Amplitude Modulation – Natural SamplingNatural 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 rd of the FET. (rd, 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 Ts.

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 – Pulse Amplitude Modulation – Flat-Top SamplingFlat-Top Sampling

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

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PAM, PWM, PPM, PCMPAM, 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 ModulationPulse Width and Pulse Position Modulation

Pulse Width ModulationPulse Width Modulation

Pulse Position ModulationPulse Position Modulation

Pulse Code Modulation (PCM)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,

fs > 2fA(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 1903.

PCM was invented by the British engineer Alec Reeves in 1937 in France.

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.

Digital ModulationDigital 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 More secure link

Digital Modulation ExampleDigital Modulation Example

The modulating signal is represented as a time-sequence of symbolsor pulses.

Each symbol has m finite states: That means each symbol carries n bitsof information where n = log2m bits/symbol.

... 0 1 2 3 T

One symbol(has m states – voltage levels)

(represents n = log2m bits of information)

Modulator

Factors that Influence Choice of Digital Factors that Influence Choice of Digital Modulation TechniquesModulation Techniques

A desired modulation scheme – Provides low bit-error rates at low SNRs

Power efficiency – Performs well in multipath and fading conditions

– Occupies minimum RF channel bandwidth 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 ModulationPower 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

PER

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Bandwidth Efficiency of ModulationBandwidth 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

bps/Hz :Efficiency BandwidthB

RB

R: the data rate (bps)B: bandwidth occupied by the modulated RF signal

Shannon’s BoundShannon’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.

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C: channel capacity (maximum data-rate) (bps)B: RF bandwidthS/N: signal-to-noise ratio (no unit)

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

Documents

aliasing slide

hz slide

sampling frequency

reconstruction theorem

sampling voice

sampling gate

harry nyquist slide

alias filter slide