1. Sampling and Aliasing

2. Review
Sampling: Time Domain
Many signals originate as continuous-time signals, e.g. conventional music or voice
By sampling a continuous-time signal at isolated, equally-spaced points in time, we obtain a sequence of numbers
k  {…, -2, -1, 0, 1, 2,…}
Ts is the sampling period. Ts t Ts
s(t)
Sampled analog waveform
impulse train

3. Sampling: Frequency Domain
Review
Sampling: Frequency Domain
Sampling replicates spectrum of continuous-time signal at integer multiples of sampling frequency
Fourier series of impulse train where ws = 2 p fs
Modulation by cos(s t)
Modulation by cos(2 s t) w
G(w) ws
2ws
-2ws
-ws
F(w)
2pfmax
-2pfmax

4. Amplitude Modulation by Cosine
Review
Amplitude Modulation by Cosine
Multiplication in time: convolution in Fourier domain
Sifting property of Dirac delta functional
Fourier transform property for modulation by a cosine

5. Amplitude Modulation by Cosine
Review
Amplitude Modulation by Cosine
Example: y(t) = f(t) cos(w0 t)
Assume f(t) is an ideal lowpass signal with bandwidth w1
Assume w1 << w0
Y(w) is real-valued if F(w) is real-valued
Demodulation: modulation then lowpass filtering
Similar derivation for modulation with sin(w0 t) w 1 w1
- w1
F(w) w
Y(w)
- w0 - w1
- w0 + w1
-w0
w0 - w1
w0 + w1 w0
½F(w - w0)
½F(w + w0)

6. Shannon Sampling Theorem
Continuous-time signal x(t) with frequencies no higher than fmax can be reconstructed from its samples x(k Ts) if samples taken at rate fs > 2 fmax
Nyquist rate = 2 fmax
Nyquist frequency = fs / 2
Example: Sampling audio signals
Human hearing is from about 20 Hz to 20 kHz
Apply lowpass filter before sampling to pass frequencies up to 20 kHz and reject high frequencies
Lowpass filter needs 10% of maximum passband frequency to roll off to zero (2 kHz rolloff in this case)
What happens if fs = 2 fmax?

7. Sampling Theorem Assumption
Continuous-time signal has no frequency content above fmax
Sampling time is exactly the same between any two samples
Sequence of numbers obtained by sampling is represented in exact precision
Conversion of sequence to continuous time is ideal
In Practice

8. Bandwidth
Bandwidth is defined as non-zero extent of spectrum in positive frequencies
Lowpass spectrum on right: bandwidth is fmax
Bandpass spectrum on right: bandwidth is f2 – f1
Definition applies to both continuous-time and discrete-time signals
Alternatives to “non-zero extent”?
Lowpass Spectrum
fmax
-fmax f
Bandpass Spectrum f1 f2 f
–f2
–f1

9. Sampled Bandpass Spectrum
Bandpass Sampling
Bandwidth: f2 – f1
Sampling rate fs must greater than analog bandwidth
For replicas of bands to be centered at origin after sampling
fcenter = ½ (f1 + f2) = k fs
Lowpass filter to extract baseband
Bandpass Spectrum f1 f2 f
–f2
–f1
Sample at fs
Sampled Bandpass Spectrum f
–f2
–f1 f1 f2

10. Sampling and Oversampling
As sampling rate increases, sampled waveform looks more like original
In some applications, e.g. touchtone decoding, frequency content matters not waveform shape
Zero crossings: frequency content of a sinusoid
Distance between two zero crossings: one half period.
With sampling theorem satisfied, a sampled sinusoid crosses zero the right number of times even though its waveform shape may be difficult to recognize
DSP First, Ch. 4, Sampling & interpolation demo

11. Aliasing Analog sinusoid Sample at Ts = 1/fs
x(t) = A cos(2p f0 t + f)
Sample at Ts = 1/fs
x[n] = x(Tsn) = A cos(2p f0 Ts n + f)
Keeping the sampling period same, sample
y(t) = A cos(2p (f0 + l fs) t + f)
where l is an integer
y[n] = y(Tsn) = A cos(2p(f0 + lfs)Tsn + f) = A cos(2pf0Tsn + 2plfsTsn + f) = A cos(2pf0Tsn + 2pln + f) = A cos(2pf0Tsn + f) = x[n]
Here, fsTs = 1
Since l is an integer, cos(x + 2 p l) = cos(x)
y[n] indistinguishable from x[n]

12. Aliasing
Since l is any integer, a countable but infinite number of sinusoids will give same sequence of samples
Frequencies f0 + l fs for l  0 are called aliases of frequency f0 with respect to fs
All aliased frequencies appear to be the same as f0 when sampled by fs

13. Folding Second source of aliasing frequencies
From negative frequency component of a sinusoid, -f0 + l fs,
where l is any integer
fs is the sampling rate
f0 is sinusoid frequency
Sampling w(t) with a sampling period of Ts = 1/fs
So w[n] = x[n] = x(Ts n)
x(t) = A cos(2  f0 t + )

14. Aliasing and Folding
Aliasing and folding of a sinusoid sin(2  finput t) sampled at fs = 2000 samples/s with finput varied
Mirror image effect about finput = ½ fs gives rise to name of folding
fs = 2000 samples/s
1000
Apparent frequency (Hz)
1000
2000
3000
4000
Input frequency, finput (Hz)

15. DSP First Demonstrations
Web site:
Aliasing and folding (Chapter 4)
Strobe demonstrations (Chapter 4)
Disk attached to a shaft rotating at 750 rpm
Keep strobe light flash rate Fs the same
Increase rotation rate Fm (positive means counter-clockwise)
Case I: Flash rate equal to rotation rate
Vector appears to stand still
When else does this phenomenon occur? Fm = l Fs
For Fm = 750 rpm, occurs at Fs = {375, 250, 187.5, …} rpm

16. Strobe Demonstrations
Tip of vector on wheel:
r is radius of disk
is initial angle (phase) of vector
Fm is initial rotation rate in rotations per second
t is time in seconds
For Fm = 720 rpm and r = 6 in, with vector initially vertical
Sample at Fs = 2 Hz (or 120 rpm), so Ts = ½ s, vector stands still:

17. Strobe Demonstrations
Sampling and aliasing
Sample p(t) at t = Ts n = n / Fs :
No aliasing will occur if Fs > 2 | Fm |
Consider Fm = Fs which could occur for any countably infinite number of Fm and Fs values:
Rotation will occur at rate of  rad/flash, which appears to go counterclockwise at rate of 0.1 rad/flash

18. Strobe Movies Fixes the strobe flash rate
Increases rotation rate of shaft linearly with time
Strobe initial keeps up with the increasing rotation rate until Fm = ½ Fs
Then, disk appears to slow down (folding)
Then, disk stops and appears to rotate in the other direction at an increasing rate (aliasing)
Then, disk appears to slow down (folding) and stop
And so forth