Sampling and Aliasing

1
Sampling and Aliasing
2
Sampling Time Domain
Review

• 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 replicates spectrum of continuous-time
  signal at integer multiples of sampling frequency
• Fourier series of impulse train where ws 2 p fs

Modulationby cos(2 ?s t)
Modulationby cos(?s t)
4
Amplitude Modulation by Cosine
Review

• 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

• Example y(t) f(t) cos(w0 t)
• Assume f(t) is an ideal lowpass signal with
  bandwidth w1
• Assume w1 ltlt 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)

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 gt 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 asnon-zero extent of
  spectrumin positive frequencies
• Lowpass spectrum on rightbandwidth is fmax
• Bandpass spectrum on rightbandwidth is f2 f1
• Definition applies to bothcontinuous-time and
  discrete-time signals
• Alternatives to non-zero extent?

9
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

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
• http//users.ece.gatech.edu/dspfirst

11
Aliasing

• Analog sinusoid
• x(t) A cos(2p f0 t f)
• Sample at Ts 1/fs
• xn 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

• yn y(Tsn) A cos(2p(f0 lfs)Tsn f) A
  cos(2pf0Tsn 2plfsTsn f) A cos(2pf0Tsn
  2pln f) A cos(2pf0Tsn f) xn
• Here, fsTs 1
• Since l is an integer,cos(x 2 p l) cos(x)
• yn indistinguishable from xn

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 ofTs 1/fs
• So wn xn 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
Apparentfrequency (Hz)
1000
2000
3000
4000
Input frequency, finput (Hz)
15
DSP First Demonstrations

• Web site http//users.ece.gatech.edu/dspfirst
• 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 gt 2 Fm
• Consider Fm -0.95 Fs which could occur for any
  countably infinite number of Fm and Fs values
• Rotation will occur at rate of -1.9 ? 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