Post on 25-Dec-2015
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Signal Processing First
Lecture 8Sampling & Aliasing
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READING ASSIGNMENTS
This Lecture: Chap 4, Sections 4-1 and 4-2
Replaces Ch 4 in DSP First, pp. 83-94
Other Reading: Recitation: Strobe Demo (Sect 4-3) Next Lecture: Chap. 4 Sects. 4-4 and 4-5
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LECTURE OBJECTIVES
SAMPLING can cause ALIASING Nyquist/Shannon Sampling Theorem Sampling Rate (fs) > 2fmax(Signal bandwidth)
Spectrum for digital signals, x[n] Normalized Frequency
22
ˆ s
s f
fT
ALIASING
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SYSTEMS Process Signals
PROCESSING GOALS: We need to change x(t) into y(t) for many
engineering applications: For example, more BASS, image deblurring,
denoising, etc
SYSTEMx(t) y(t)
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System IMPLEMENTATION
DIGITAL/MICROPROCESSOR Convert x(t) to numbers stored in memory
ELECTRONICSx(t) y(t)
COMPUTER D-to-AA-to-Dx(t) y(t)y[n]x[n]
ANALOG/ELECTRONIC: Circuits: resistors, capacitors, op-amps
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SAMPLING x(t)
SAMPLING PROCESS Convert x(t) to numbers x[n] “n” is an integer; x[n] is a sequence of values Think of “n” as the storage address in memory
UNIFORM SAMPLING at t = nTs
IDEAL: x[n] = x(nTs)
A-to-Dx(t) x[n]
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SAMPLING RATE, fs
SAMPLING RATE (fs) fs =1/Ts
NUMBER of SAMPLES PER SECOND
Ts = 125 microsec fs = 8000 samples/sec• UNITS ARE HERTZ: 8000 Hz
UNIFORM SAMPLING at t = nTs = n/fs
IDEAL: x[n] = x(nTs)=x(n/fs)
A-to-Dx(t) x[n]=x(nTs)
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Hz100f
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SAMPLING THEOREM
HOW OFTEN ? DEPENDS on FREQUENCY of SINUSOID ANSWERED by NYQUIST/SHANNON Theorem ALSO DEPENDS on “RECONSTRUCTION”
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Reconstruction? Which One?
)4.0cos(][ nnx )4.2cos()4.0cos(
integer an is Whennn
n
Given the samples, draw a sinusoid through the values
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STORING DIGITAL SOUND
x[n] is a SAMPLED SINUSOID A list of numbers stored in memory
EXAMPLE: audio CD CD rate is 44,100 samples per second
16-bit samples Stereo uses 2 channels
Number of bytes for 1 minute is 2 X (16/8) X 60 X 44100 = 10.584 Mbytes
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sfsT
nAnx
ˆ
)ˆcos(][
)cos()(][)cos()(
ss nTAnTxnxtAtx
DISCRETE-TIME SINUSOID
Change x(t) into x[n] DERIVATION
))cos((][ nTAnx s
DEFINE DIGITAL FREQUENCY
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DIGITAL FREQUENCY
VARIES from 0 to 2, as f varies from 0 to the sampling frequency
UNITS are radians, not rad/sec DIGITAL FREQUENCY is NORMALIZED
ss f
fT
2ˆ
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SPECTRUM (DIGITAL)
sf
f 2ˆ
kHz1sf 2–2
))1000/)(100(2cos(][ nAnx
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SPECTRUM (DIGITAL) ???
2–2
?
x[n] is zero frequency???
))100/)(100(2cos(][ nAnx
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The REST of the STORY
Spectrum of x[n] has more than one line for each complex exponential Called ALIASING MANY SPECTRAL LINES
SPECTRUM is PERIODIC with period = 2 Because
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ALIASING DERIVATION
Other Frequencies give the sameHz1000at sampled)400cos()(1 sfttx
)4.0cos()400cos(][ 10001 nnx n
Hz1000at sampled)2400cos()(2 sfttx
)4.2cos()2400cos(][ 10002 nnx n
)4.0cos()24.0cos()4.2cos(][2 nnnnnx
][][ 12 nxnx )1000(24002400
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ALIASING DERIVATION–2
Other Frequencies give the same
ss f
fT
2ˆ 2
s
s
ss
s
f
f
f
f
f
ff 22)(2ˆ :then
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ALIASING CONCLUSIONS
ADDING fs or 2fs or –fs to the FREQ of x(t) gives exactly the same x[n] The samples, x[n] = x(n/ fs ) are EXACTLY
THE SAME VALUES
GIVEN x[n], WE CAN’T DISTINGUISH fo
FROM (fo + fs ) or (fo + 2fs )
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NORMALIZED FREQUENCY
DIGITAL FREQUENCY
ss f
fT
2ˆ 2
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SPECTRUM for x[n]
PLOT versus NORMALIZED FREQUENCY INCLUDE ALL SPECTRUM LINES
ALIASES ADD MULTIPLES of 2 SUBTRACT MULTIPLES of 2
FOLDED ALIASES (to be discussed later) ALIASES of NEGATIVE FREQS
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SPECTRUM (MORE LINES)
2–0.2 1.8–1.8
))1000/)(100(2cos(][ nAnx
kHz1sf
sf
f 2ˆ
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SPECTRUM (ALIASING CASE)
–0.5–1.5 0.5 2.5–2.5 1.5
))80/)(100(2cos(][ nAnx
sf
f 2ˆ
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SAMPLING GUI (con2dis)
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SPECTRUM (FOLDING CASE)
0.4–0.4 1.6–1.6
))125/)(100(2cos(][ nAnx