1. Lecture 4 Sampling & Aliasing
Soutenance de thèse vendredi 24 novembre 2006, Lorient
Lecture 4
Sampling & Aliasing

2. Classification of Signal
Continuous-time and discrete-time signal
Analog and digital signal (time and amplitude)
Continuous-time signal
Discrete-time signal：
Discrete variableContinuous amplitude
Time-domain discrete signals
Analog Signal: Continuous variableContinuous in time and amplitude,
Example: voltage, current, temperature,…
Digital Signal：Discrete variablesDiscrete both in time and amplitude

3. Transform, filter, inspection, spectrum analysis;
Signal Processing
Representation, transformation and manipulation
of signals and the information they contain.
Signal operation include:
Transform, filter, inspection, spectrum analysis;
Modulation and coding;
Analog Signal Processing;
Digital Signal Processing.

4. DSP、FPGA、SOPC、SOC、Algorithm Codes
Signals and Systems
Basic model:
Input:x Output: y
System: h
DSP、FPGA、SOPC、SOC、Algorithm Codes

5. Three Problems h Given x and h, find y analysis
x y h
Given x and h, find y analysis
Given h and y, find x control
Given x and y, find h design or synthesis

6. Processing of analog signal with digital methods
Digitalized process for analog signals
Sample
Quantizer
Coder
xa(t)
x(n)
Digital processing method
A/D
DSP
D/A
xa(t)
ya(t)
Filter
x(n)
y(n)

7. Discrete-time signals
A continuous signal input is denoted x(t)
A discrete-time signal is denoted x(n), where n = 0, 1, 2, …
Therefore a discrete time signal is just a collection of samples obtained at regular intervals (sampling frequency)

8. UNIFORM SAMPLING at t = nTs
SAMPLING x(t)
SAMPLING PROCESS
Convert x(t) to numbers x[n]
“n” is an integer; x[n] is a sequence of values
UNIFORM SAMPLING at t = nTs
IDEAL: x[n] = x(nTs)
C-to-D
x(t)
x[n]

9. UNIFORM SAMPLING at t = nTs = n/fs
SAMPLING RATE, fs
SAMPLING RATE (fs)
fs =1/Ts
NUMBER of SAMPLES PER SECOND
Ts = 125 microsec  fs = 8000 samples/sec
UNITS ARE HERTZ: Hz
UNIFORM SAMPLING at t = nTs = n/fs
IDEAL: x[n] = x(nTs)=x(n/fs)
C-to-D
x(t)
x[n]=x(nTs)

10. Periodic (Uniform) Sampling
Sampling is a continuous to discrete-time conversion
Most common sampling is periodic
T is the sampling period in second
fs = 1/T is the sampling frequency in Hz
Sampling frequency in radian-per-second s=2fs rad/sec
Use [.] for discrete-time and (.) for continuous time signals
This is the ideal case not the practical but close enough -3 -2 -1 1 2 3 4

11. The Sampling Theorem & Impulse-Train Sampling
-3T -2T -T T T 3T T t

12. Representation of Sampling
Mathematically convenient to represent in two stages
Impulse train modulator
Conversion of impulse train to a sequence
s(t)
Convert impulse train to discrete-time sequence
xc(t) x
x[n]=xc(nT)
xc(t)
x[n]
s(t) t n
-3T
-2T -T T 2T 3T 4T -3 -2 -1 1 2 3 4

14. SAMPLING THEOREM HOW OFTEN ? DEPENDS on FREQUENCY of SINUSOID
ANSWERED by SHANNON/NYQUIST Theorem
ALSO DEPENDS on “RECONSTRUCTION”

15. Periodic Sampling Sampling is, in general, not reversible
Given a sampled signal one could fit infinite continuous signals through the samples 20 40 60 80
100
-0.5
0.5 1 -1
Fundamental issue in digital signal processing
If we loose information during sampling we cannot recover it
Under certain conditions an analog signal can be sampled without loss so that it can be reconstructed perfectly

16. DISCRETE-TIME SINUSOID
Change x(t) into x[n] DERIVATION
DEFINE DIGITAL FREQUENCY

17. VARIES from 0 to 2p, as f varies from 0 to the sampling frequency
DIGITAL FREQUENCY
VARIES from 0 to 2p, as f varies from 0 to the sampling frequency
UNITS are radians, not rad/sec
DIGITAL FREQUENCY is NORMALIZED

18. SPECTRUM (DIGITAL)
–0.2p
2p(0.1)

19. Spectrum of x[n] has more than one line for each complex exponential
SPECTRUM (DIGITAL)
Spectrum of x[n] has more than one line for each complex exponential
Called ALIASING
MANY SPECTRAL LINES
SPECTRUM is PERIODIC with period = 2p
Because

20. Other Frequencies give the same
ALIASING DERIVATION
Other Frequencies give the same

21. Other Frequencies give the same
ALIASING DERIVATION–2
Other Frequencies give the same

22. GIVEN x[n], WE CAN’T DISTINGUISH fo FROM (fo + fs ) or (fo + 2fs )
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 )

23. NORMALIZED FREQUENCY
DIGITAL FREQUENCY

24. SPECTRUM (MORE LINES)
2p(0.1)
–0.2p
–1.8p
1.8p

25. SPECTRUM (ALIASING CASE)