Chapter 6: Sampling and pulse modulation A. Bruce Carlson ...bazuinb/ECE4600/Ch06_02.pdf ·...

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Communication Systems, 5e

Chapter 6: Sampling and pulse modulation

A. Bruce CarlsonPaul B. Crilly

Chapter 6: Sampling and pulse modulation

• Sampling theory and practice• Pulse-amplitude modulation• Pulse-time modulation

Sampling Theory and Practice

• The spectrum of a sampled signal– The time domain and spectrum of a sampling waveform

• Minimum sampling frequency– Based on the maximum allowable aliasing error,

message BW, LPF characteristics, etc.– The Nyquist Rate

• Practical sampling versus ideal sampling• Signal reconstruction• PAM, PDM and PPM

Sampling

• Multiplicative, periodic sampling waveform tptxtxs

tpFtxFtxF s

tpTnttp pulsen

tp pulsen ss

sT 2sT

tppulse

Convolution with Sampling Spectrum

• Spectral replication of F[x(t)]– If x(t) not band limited, there will be spectral aliasing– The shape of the sampling pulse may change the

magnitude and phase of the spectral replicas!

tpFtxFtxF s

n sspulses T

tptxtx 11

Spectral Replication

• Reduce the sampling rate to the minimum txtstxs

tXfStXs

sffreq

W sf2sfsf2 W

Replication Interval

sffreq

W sf2sfsf2 W

Figure 6.1-3, Spectra for switching sampling (a) message (b) sampled message (c) sampled message

Nyquist Rate

WWf min,s

Aliasing by an ADC

• The desired baseband signal spectrum prior to sampling, typical definitions– Filter Design for baseband signals or complex signals– Center frequency is zero

Practical Filter Construction

Wffff passbandtransitionpassbands 2

• For practical applications, we usually sample at greater than the Nyquist rate. This allows for a guard band around the signal of interest (SOI)

• If ADC systems are involved, many times we use 4 x W sampling, 2 times the Nyquist rate or even higher.

Wfff stopbandpassbands 2

Wf passband

stopbandtransitionpassband fff

Bandpass Aliasing by an ADC• The desired baseband signal spectrum prior to

sampling.– Filter design for bandpass signal sampling.– Signal-of-interest center frequency is fs/4

Sampling in Matlab

• Using the interpolated message from before …• A sampling rate of fs/4 can be used

0 1 2 3 4 5 6

0Sequenctial FFTs of the message

Frequency (Hz)

0 2 4 6 8 10 12 14

-20Sequenctial FFTs of the Sampled Waveform

Frequency (Hz)

Question, what if I wanted to shift a baseband signal to a higher frequency … sample(?) and BPF!

Aliasing (1)

• The frequency domain response of the perfect sampling function is:

• Convolve with the input signal spectrum …

sF sF2 sF3 sFsF3 sF2

k2jjT2TnjexpjP

Aliasing (2)

• The frequency bands that “could” be aliased when sampled are

p jkjGT1

Tk2jjG

Aliasing Example (1)

• Predict the aliasing result for

sF2 sF2Fs

Aliasing Example (2)

Original spectrum

sF2 sF2Fs

f2F3 ss

f2F ss

2F3 ss

Aliased Baseband Spectrum

Interpolation-Filter Example

• 2-tone test signal (30&60 Hz, fs = 1000 Hz)– fft scaled to maintain power

• interpolation x4 (upsample function)– scaled by 4 so interpolated signal has the same power

• interp-filter (interp function)– filter includes “interpolation gain” of x4

Continuous Time Reconstruction

• Reconstruction of a discrete time waveform into a continuous time waveform.– If it’s digitized, it probably needs to be restored to an analog

waveform at sometime.

• The sampled representation– Shown as a spectrum from –fs/2 to + fs/2

or for DSP people from 0 to fs (it’s shown aliased!?)– It theoretically contains all frequency replicas!

p jkjGT1

Tk2jjG

Continuous Time Reconstruction

• Impulse outputs at the “sample times”• Perfect Reconstruction Filter to eliminate replicas

– Derived from rect in the frequency domain– The convolution of the samples with a sinc function

delayttBBKth 2sinc2

delaytf2jxpeB2frectKfH

nTtnTxtx delay

nsincˆ

thtxtx̂ s

BKLet 21

Reconstruction (2)

• Perfect Reconstruction Filter– Derived from rect in the frequency domain– The convolution of the samples with a sinc function

sincnTtnTxtx̂ delay

sincnTxtx̂ delay

0delaytLet

nTtnTxtx

nsincˆ

Reconstruction (3)

• Perfect Reconstruction Filter– Each sample causes a time offset sinc– All the sinc’s are summed to form the continuous signal

nTtnTxtx

nsincˆ

-5 -4 -3 -2 -1 0 1 2 3 4 5

Individual sinc functions with T delays

Reconstruction (4)

• Perfect Reconstruction Filter

nTtnTxtx

nsincˆ

-80 -60 -40 -20 0 20 40 60 80-0.4

-80 -60 -40 -20 0 20 40 60 80-0.2

Individual sinc’s Resulting summation

Alternate Functions Used for Reconstruction

• Zero Order Hold (like a Digital to analog converter)– Should be followed by LPF

• First Order Hold (Triangle)

TnTtectrnTxtx̂

TnTttrinTxtx̂

Signal reconstruction from sample (a) ZOH

Figure 6.1-8

-80 -60 -40 -20 0 20 40 60 800

Signal reconstruction from sample (b) FOH

Figure 6.1-8

-80 -60 -40 -20 0 20 40 60 800

Aliasing Error in Reconstruction

• Estimated as the amount of filter power that comes from the first adjacent spectral replica.– 1st order Butterworth LPF and spectral replica– Integrate the aliased portion of the filter in the passband– 1st order estimate: the aliasing filter power at the passband bandedge

Figure 6.1-9

(a) output of RC filter, (b) after sampling

Butterworth LPF

Aliasing Computation

• Dr. Bazuin’s assumptions:– Passband has negligible

attenuation (book use W with B at the 3 dB point)

– Aliasing at passband W not B (book uses B)

– A digital filter will be used to “clean-up” the transition band.

Figure 6.1-9: (a) output of RC filter, (b) after sampling

voltageVoltage

errorAtten2

error 2

Telephony Example

• Maximum voice W=3.4 kHz, find the sample rate– Passband 0.5 dB ripple, N=4th order Butterworth filter– Find the filter cutoff (3 dB) frequency– Stopband Attenuation -40 dB, find the stopband frequency– Compute the sample rate

7556.0,7687.0 Let

kHzkHzkHzB 423.47556.04.34.3

16.3,16.3 Let

kHzkHzWfB

s 985.1316.3423.4

kHzkHzkHzfs 385.174.3985.13

Telephony Example (2)• Maximum voice 3.4 kHz, find the sample rate

– Passband 0.5 dB ripple, N=1st order Butterworth filter– Find the filter cutoff (3 dB) frequency– Stopband Attenuation -40 dB, find the stopband frequency– Compute the sample rate

3487.0,3493.0 Let

kHzkHzkHzB 733.93487.04.34.3

995.99,995.99 Let

kHzkHz

kHzBWfs

295.973995.99733.975.9

kHzkHzkHzBWfs 695.9764.3295.973

Pulse-Based Modulation

• As long as sampling is performed at appropriate sample rates, any communications signal that conveys the sampled value during the sample time interval can communicate a continuous waveform.– The carrier doesn’t have to be continuous, it can be

different … as long as the sample value can be recovered.

• Pulse Communications– Pulse with amplitude (PAM)– Pulse with duration (PDM or PWM)– Pulse with a position in the sample time frame (PPM)

Pulse-Amplitude Modulation

• The pulse output from an instantaneous sampler

nTtpnTxtxn

TnttpTntp

TntnTxtpTnttpnTxtxnn

txtptxPAM

fXfPfXPAM

Aperture Effects result from the “time aperture” p(t)

PAM Pulses

• For unipolar signals the pulses may appear as an AM modulation amplitude– AM with pulses instead of a carrier

tm1txp

nTtpnTm1txn

Analog signal and corresponding PAM signal

Figure 6.2-1

(a) sample & hold circuit (b) waveformsFigure 6.2-2

Flat-top sampling

• PAM time delayed by ½ the PAM width– If transmitted, does the time delay matter? …

… probably not

PAM Waveforms

• The modulation is dependent upon the pulse amplitude.– Signals near zero may be hard to detect– Can the receiver detect positive and negative pulses– Therefore, use an AM like offset for the amplitude

“Called uni-polar flat top in the text”

ssp TktpTkxAtx 10

01 sTkxwhere

PAM Spectral Content

• Convolution of the pulse and the AM-like waveform

• Spectrum based on “widest: frequency element, typically the symbol period or sample pulse.– Use sinc null-to-null as Bandpass Bandwidth, BT

tpFtxFtxF s

fMfAfXtxFtmAtx 00 1

sn ss Tc

TtpF sin11

21 WBT

PAM Applications

• Rarely used for single channel communication systems, but …

• used in conjunction with instrumentation, data telemetry, and instrumentation systems

• One element of aTime-division multiplexing (TDM) systems

• A basis for other digital modulation systems

Pulse-Time Modulation

• PAM receivers require amplitude to be determined for brief pulses, can we translate the sampled signals into a form that might be easier to receive?– One not dependent upon amplitude?

• Pulse-Duration Modulation , PDM(also called Pulse-Width Modulation, PWM)– The length of the pulse width– Nominally centered on the periodicity

• Pulse-Position Modulation, PPM– The position of the pulse relative to the pulse period– Pulse widths are fixed

Figure 6.3-1

Types of pulse-time modulation

(a) Generation of PDM and PM signals, (b) waveforms

Generation of PWM and PPM

PWM and PPM Bandwidth

• Based on the minimum pulse width, but …– The value is dependent upon accurately measuring

time. Therefore, the faster the receiver rise-time in tracking the signal, the more accurate the analog measurement.

– Note 1: For PPM, if the pulse locations are describe using probability, the power spectral density can be computed as the product of the PSD and sinc.

– Note 2: For PWM, the pulse center and width should both be described probabilistically.

WTt sr

Conversion of PDM or PPM into PAM for Demodulation

Figure 6.3-3

Desired PAM Amplitude

Chapter 6: Sampling and pulse modulation A. Bruce Carlson ...bazuinb/ECE4600/Ch06_02.pdf ·...

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