Chapter 6: Sampling and pulse modulation A. Bruce Carlson ...bazuinb/ECE4600/Ch06_01.pdf · Chapter...
Transcript of Chapter 6: Sampling and pulse modulation A. Bruce Carlson ...bazuinb/ECE4600/Ch06_01.pdf · Chapter...
Communication Systems, 5e
Chapter 6: Sampling and pulse modulation
A. Bruce CarlsonPaul B. Crilly
© 2010 The McGraw-Hill Companies
Chapter 6: Sampling and pulse modulation
• Sampling theory and practice– Basis for digital signal processing
• Pulse-amplitude modulation• Pulse-time modulation
© 2010 The McGraw-Hill Companies
Sampling Theory and Practice
• The spectrum of a sampled signal– The time domain and spectrum of a sampling waveform
(periodic square and pulse signals from Chap. 2)
• 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
spulse
ns T
trecttpTnttp
sspulsen ss
TfTtpT
nfT
tp
sinc11
sT2sT
sT 2sT
tppulse
p(t) sampling waveform
Sampling Waveforms
• Unit level sampling
• Narrow rect function (mult in time, conv. in freq.)
• “Time limited” sampling function both frequency and time can be derived. (Cos pulse)
ffPtp pulsepulse 1
sss
TfTfPTtrecttp
sinc
ccpulsecpulse fffffPtftp 2cos
scsscs TffTTffTfP sincsinc
Sampling Waveforms
• Impulse– Pulse as an impulse … FT is impulses spaced at 1/Ts
• Sampling signal spectrum
• Sampled Signal Spectrum (spectral replication!)
1 fPttp
n ssn ss Tnf
TTnf
TfPtp 11111
n sss T
nfT
txFtxF 11
n
sTnttp
Sampling Waveforms
• Narrow rect– Pulse as a narrow rect function … “scaled impulses”
• Sampling signal spectrum
• Sampled Signal Spectrum (weighted spec. replicas!)
ffPtrecttp pulsepulse sinc
n ssn ss Tnff
Tf
Tnf
TtpF 1sincsinc11
n ssss T
nfTn
TtxFtxF 1sinc1
Sampling Waveforms
• Wide rect– Pulse as a wide rect function … filling the sampling
period (a degenerate case, not practical but …)– If you draw this it is a constant “1”
• Sampling signal spectrum
• Sampled Signal Spectrum (degenerate case)
sspulses
pulse TfTfPTtrecttp
sinc
txFftxFtxF s
n ss
sss
n ss Tnfn
TTTfT
Tnf
TtpF 1sincsinc11
fTfTT
nfT
tpF ssn ss
sinc11
Convolution with Sampling Spectrum
• Spectral replication of F[x(t)]– If x(t) is not band limited, there will be an overlapping
of the adjacent impulses specrtrum– If sampling pulse not “appropriate” some of the replicas
will be smaller than others … periodically (narrow rect pulse example or the “pulses” spectral envelope)
tpFtxFtxF s
Replicas and Spectral Overlap
• As seen, the spectral replicas occur at a spacing of 1/Ts or fs, the sampling rate.
• To avoid any chance of overlap, the sampled signal must be band limited to:
• If you want to pick a sampling rate based on W
s
s
TfW
21
2
Wfs 2W
Ts
21
Nyquist Sampling Rate
• The Nyquist sampling rate defines the minimum sampling frequency for “distinct” replicated spectrum.– It also means you should be able to reconstruct the
original continuous time signal
• Aliasing is when spectral replicas overlap into adjacent spectral regions– occurs when there is an insufficient guard band– Remember the filter transition bands,
there will always be some aliasing, but are the aliased signal level sufficiently attenuated (at the stopband level)
Bandpass SamplingAdvanced Concept
• If a complex signal to be sampled is band limited
• What will happen after ideal (impulse) sampling?
• This also works for real signal with
22
ss
ss
ffnfXffn
2
sss
ffnfXfn
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(a) functional (b) waveforms (c) circuit
Switching sampler
tstxtxs
See LTSpice
Chopper Sampling
• Sampling function– The mathematical description
– The Fourier Series of repeated rects
n
Tnttrectts
2Tt2
Tfor,trectts
n
00n T1ffor,tfn2jexpcts
2
T
2T
0n dttfn2jexptsT1c
Chopper Sampling (2)
• Sampling function Fourier series coefficients
n
Tnttrectts
2
20
02
2
0n fn2jtfn2jexp
T1dttfn2jexp
T1c
0
0
0
0n fn2j
2fn2jexp
fn2j2fn2jexp
T1c
0
0
0
0
0n fnsinc
Tfnfnsin
T1
fn2j2fn2sin2j
T1c
Chopper Sampling (3)
• Sampling function Fourier series coefficients
n
Tnttrectts
0sinc fnT
cn
1n
0tfn2jtfn2j
n0 T1ffor,eeccts 00
1n
00n0 T1ffor,tfn2cosc2cts
1n
000n0 T1ffor,fnffnfcfcfS
n T
fforfnffnT
fS 1,sinc 000
Chopper Sampling (4)
• Time waveform with Fourier coefficients txtstxs
1n
000n0s T1ffor,fnfXfnfXcfXcfX
1n
00n0s T1ffor,tfn2costxc2txctx
• Spectral replicates at a spacing of fo=1/T– For unity gain, the chopper should have gain 1/co
Spectra for switching sampling
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(a) message (b) sampled message (c) sampled message 2sf W 2sf W
Replica envelope due to sampling pulse shape
Aliasing/overlapping due to signal bandwidth
Nyquist Sampling Rate
• Reduce the sampling rate to the minimum
WWfs min,
Wfs 2min,
ss T
f 1
WTs
21sT
W
21
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Practical Filter Construction
W2fs
• 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 DSP systems are involved, many times we use 4x to 8x sampling, 2 to 4 times the Nyquist rate.
Filter Reminder
• Passband– Frequencies where signal is
meant to pass
• Stopband– Frequencies where some defined
level of attenuation is desired
• Transition-band– The transitions frequencies
between the passband and the stopband
PBBW
SBBW
passbandstopbandpracticals fff ,
Minimum Filter Construction
Wfff stopbandpassbands 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 (For the 1st order derivative approx to be valid may use – 5x to 8x)
sfWsf WWfs Wfs
stopbandpassbands fff Wf passband Wff sstopband
Aliasing by an ADC
• The desired baseband signal spectrum prior to sampling.
sF2Fs
sF2Fs
Aliasing (2)
• The frequency domain response of the perfect sampling function is:
• Convolve with the input signal spectrum …
sF sF2 sF3 sFsF3 sF2
kn T
k2jjT2TnjexpjP
0
Aliasing (3)
• The frequency bands that “could” be aliased when sampled are
sF sF2 sF3 sFsF3 sF2
0
k
Tk
p jkjGT1
Tk2jjG
T1jG
Aliasing Example (1)
• Predict the aliasing result for
sF2Fs
2F3 s
sF2 sF2Fs
Aliasing Example (2)
Original spectrum
sF2Fs
2F3 s
sF2 sF2Fs
2Fs
2Fs
2F
f2F3 ss
2F
f2F ss
2F3
f2F ss
2F5f
2F3 ss
2Fs
2Fs 0
Aliased Baseband Spectrum
Ideal Sampling
• Remembering delta function approximations from Chapter 2
nn0
TntTnttrect1ts lim
txtstx
nnTntTnxTnttxtx
sXfSfX
n T1nfX
T1fX
Sampling in Matlab
• Using the 16x interpolated message from before …• A sampling rate of fs/4 can be used
0 1 2 3 4 5 6
x 104
-150
-100
-50
0Sequenctial FFTs of the message
Frequency (Hz)
Pow
er (d
B)
0 2 4 6 8 10 12 14
x 104
-160
-140
-120
-100
-80
-60
-40
-20Sequenctial FFTs of the Sampled Waveform
Frequency (Hz)
Pow
er (d
B)
Question, what if I wanted to shift a baseband signal to a higher frequency …
Sampling in Matlab
0 1 2 3 4 5 6
x 104
-150
-100
-50
0Sequenctial FFTs of the message
Frequency (Hz)
Pow
er (d
B)
0 2 4 6 8 10 12 14
x 104
-160
-140
-120
-100
-80
-60
-40
-20Sequenctial FFTs of the Sampled Waveform
Frequency (Hz)
Pow
er (d
B)
… sample and BPF ?!
Question, what if I wanted to shift a baseband signal to a higher frequency …
This is called interpolation-filtering (ECE 6560)
Subsampling Theorem
• Also called bandpass sampling …• What would happen if a had a sampling waveform that
was an integer fraction of the carrier? (i.e. fs=fc/4)– Spectral comb function goes to infinite at nfs !!– Convolve in the frequency domain with a complex
narrowband bandpass signal …..
– Where do the replicas appear … at all the combs, even zero– Apply a low pass filter … and you have downconverted the
complex RF to Baseband! (who needs an RF mixers !)– There are numerous commercial ADCs that allow this to
occur.
Bipolar Chopper: Example 6-1Square wave mixing!
• Using the Fourier Series of a Square Wave
nTnt
Ttrect
Ttrectts2
2
tf52cos54tf32cos
34tf2cos4ts 000
• Generating odd sample rate spectrum
00
00
00
f5ff5f52
f3ff3f32
ffff2fS
Bipolar Chopper (2)
• Using the Fourier Transform
n T
nfT
TfTTfTtsF 11sinc2
sinc2
2
nTnt
Ttrect
Ttrectts2
2
n T
nfTfTftsF 1sinc2
sinc
n TnffntsF 1
2sinc
• The odd harmonics of the period are scaled by sinc(n/2),all others are zero … now convolve with the signal!
Bipolar Chopper (3)
• Using the Fourier Transform
Tf
Tf
TftsF 5
25sinc3
23sinc1
21sinc
Tf
Tf
TftsF 5
523
3212
t
Tt
Tt
Tts 52cos
5432cos
3412cos4
• Same result as Fourier Series for a square wave!• math should work both ways …
Bipolar Chopper Application
• Chopper-Stabilized Amplifier– Chop the input signal– Filter and apply gain to bandpass signal centered at fs
– Chop to return the signal to DC– Filter to keep just the low pass signal
• A technique used for the first low frequency analog MOSFET amplifiers due to 1/f noise.
Reconstruction
• Reconstruction of a discrete time waveform into a continuous time waveform needs an appropriate filter!
• Perfect Reconstruction Filter– Derived from rect (perfect passband) in the frequency domain– The convolution of the samples with a sinc function
delayttBBKth 2sinc2
delaytf2jxpeB2frectKfH
Ttt
nTtnTxtx delay
nsincˆ
thtxtx̂ s
BKLet 21
Reconstruction (2)
• Perfect Reconstruction Filter– Derived from a rect in the frequency domain– The convolution of the samples with a sinc function
Ttt
sincnTtnTxtx̂ delay
n
TtnTt
sincnTxtx̂ delay
n
0delaytLet
nTtnTxtx
nsincˆ
-5 -4 -3 -2 -1 0 1 2 3 4 5
-0.2
0
0.2
0.4
0.6
0.8
1
Reconstruction (3)
• Perfect Reconstruction Filter
nTtnTxtx
nsincˆ
-80 -60 -40 -20 0 20 40 60 80-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-80 -60 -40 -20 0 20 40 60 80-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Alternate Functions Used for Reconstruction
• Zero Order Hold– Should be followed by a LPF
• First Order Hold (Triangle)
TnTtectrnTxtx̂
nrect
TnTttrinTxtx̂
nrect
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Signal reconstruction from sample (a) ZOH
Figure 6.1-8
ZOH
-80 -60 -40 -20 0 20 40 60 800
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-80 -60 -40 -20 0 20 40 60 800
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
sTnf 1
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Signal reconstruction from sample (b) FOH
Figure 6.1-8
FOH
-80 -60 -40 -20 0 20 40 60 800
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-80 -60 -40 -20 0 20 40 60 800
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1