MIT OpenCourseWare
http://ocw.mit.edu
MAS.160 / MAS.510 / MAS.511 Signals, Systems and Information for Media Technology
Fall 2007
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Shannon Sampling Theorem
Sampling
f s > 2 f max
x ( t ) = cos(2 1500t)
f s = 2 f max
t = Tsn
fo =
1
x [ n ] = cos(2 �1500 � Tsn )
t = Tsn
f max < f s 2
Ts =
Nyquist rate
1
fs
1500 Hz
0.5
f s = 8000 Hz
Ts =
1
= 8000
sec
1
fs
fs = 8000Hz
0
-0.5
1
x [ n ] = cos(2 �1500 � 8000
n)
f o
=
1
-1
0
0.2
0.4
0.6
0.8
1
t - sec
1.2
1.4
1.6
1.8
2
x 10-3
oversampling
1
1500 Hz
0.8
2A
0.5
0.6
0.4
0
0.2
� f max
2
-0.5
0
-1
0
0.2
0.4
0.6
0.8
1
t - sec
1.2
1.4
1.6
1.8
-1
x 10
-0.5
� fs
2
� f2s
Time domain
0.9
0.6
0.8
0.4
x
0.2
1
fs
x 104
sampled sinewave
sampling
0.6
0.5
-0.2
0.4
-0.4
0.3
-0.6
0.2
1500 Hz
1
0.5
0
0.1
0
0.2
0.4
0.6
0.8
1
t - sec
1.2
1.4
1.6
1.8
0
2
x 10
-3
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
x 10-4
Ts
0
2Ts
3Ts
-0.5
4Ts
Frequency domain
1
1
0.9
0.9
0.8
0.8
0.7
-1
0
0.2
0.4
0.6
0.8
1
t - sec
1.2
1.4
1.6
1.8
2
x 10-3
oversampling
1
� fo
0.8
fo
0.7
0.6
0.6
0.6
2A
0.5
comb
0.5
0.4
0.4
0.3
� fs + fo
2A
0.4
� f2s
0.2
0.3
fs � fo
fs + fo
� fs � fo
0.2
fs
2
0.2
0
0.1
0.1
0
fs
2
0.7
0
-0.8
2A
fo
Hz
1
0.8
-1
0
f - freq
-3
1500 Hz
1
� fo
f max
2
0.5
-1
-0.5
0
f/fs - freq
� fo
0.5
1
x 10
fo
4
0
-2
-1.5
�2 f s
-1
-0.5
� fs
0
0.5
1
fs
1.5
2
x 104
2 f s
-1
-0.5
0
ff - freq
Hz
0.5
1
x 104
sinewave
spectrum
copied
around
each comb
“tooth”
D-to-C
only reconstructs
frequencies
between
�
fs
2
� f �
fs
2
Normalized frequency
Sampling: Folding
x[n] = x(nTs ) = cos(2f o nTs ) = cos(� o nTs )
f̂ = �ˆ (2 ) = fTs = f f s
�ˆ = �Ts
ˆ ) = cos(n�ˆ )
x[n] = cos(2fn
fo =
1
fs
2
< fo < fs
0.5 < f̂ o < 1
1500 Hz
fo =
1
f s = 8000 Hz
5600 Hz
f o �, f apparent �
0.5
0.5
f apparent = f s � f o
0
-0.5
-1
0
0.2
0.4
0.6
0.8
1.4
�1+ f o f s
0.6
0.4
1.2
1.6
1.8
2
x 10-3
waveform spectrum
copies centered
at integer values
of f̂
f̂ = f o f s
� fo fs
0.8
2A
1
t - sec
oversampling
1
o +freq
x -freq
R -copy
B orig
G +copy
0
frequencies
divided by
sampling rate
1� f o f s
�1� f o f s
o +freq
x -freq
K 2nd - copy
R 1st -copy
B orig
G 1st +copy
K 2nd +copy
2A
-1
= 2400Hz
0
0.2
0.4
0.6
0.8
1
t - sec
�1+ f o f s
0.8
0.4
1.4
1.6
1.8
2
x 10-3
folding
1
0.6
1.2
f̂ Bo >
1� f o f s
� fo fs
fo fs
2 � fo fs
�2 + f o f s
reconstructed
-1
-0.5
0
0.5
f̂f - freq
1
f̂ max <
0 > f̂Gx >
0.2
Shannon Sampling
Theorem
1.5
1
2
f̂ Bx < � 12
1+ f o f s
0.2
0
-1.5
= 8000 � 5600Hz
-0.5
0
-1.5
-1
-0.5
0
0.5
f̂f - freq
1
2
Sampling: Aliasing
1
1.5
� < f̂ Ro < 0
x >o
" flipped"
Sampling:
f s � f o < 1.5 f s
1 � f̂ o < 1.5
f o = 8700 Hz
1
f s = 8000 Hz
f apparent = f o � f s
0
= 8700 � 8000Hz
= 800Hz
-0.5
o +freq
x -freq
K 2nd - copy
R 1st -copy
B orig
G 1st +copy
K 2nd +copy
2A
-1
0
0.2
0.4
0.6
0.8
1
t - sec
1.2
1.4
1.6
1.8
2
x 10-3
apparent frequency
f o �, f apparent �
0.5
fs
2
…
aliasing
1
�1+ f o f s
0.8
oversampled fold
0
fs
f̂ Bo > 1
1� f o f s
f̂ Bx < �1
alias
fs
2
0.6
2 � fo fs
0.4
0.2
0
-1.5
� fo fs
-1
-0.5
0
f - freq
0.5
reconstructed
fo fs
�2 + f o f s
1
0 > f̂ Ro >
1.5
1
2
� 12
< f̂Gx < 0
o> x
1
2
1
2
input frequency fo
freqramp
Vary Sample rate
Vary Sample rate
x(t ) = cos(2 1200t )
t = Tsn
x(t ) = cos(2 1200t )
t = Tsn
var y Ts
x[n] = cos(2 �1200 � Ts n)
fs=4200 Hz,
1
fo=1200Hz
folding
fo=1200Hz
0.5
0
0
-0.5
-0.5
0
0.2
0.4
0.6
0.8
1
t - sec
1.2
1.4
1.6
1.8
-1
2
x 10-3
0
0.2
0.4
0.6
0.8
oversampling
1
1
t - sec
1.2
1.4
1.6
1.8
2
x 10-3
folding
1
0.8
0.8
0.6
2A
fs=1900 Hz,
1
0.5
-1
var y Ts
x[n] = cos(2 �1200 � Ts n)
�
fs
2
fs
2
� f2s
0.6
fs
2
2A
0.4
0.4
0.2
0.2
0
-6000
-4000
-2000
0
f/fsf- freq
2000
4000
0
6000
-2500
-2000
Sampling: Composite Signal
fs=3800 Hz,
-1000
-500
0
500
f/fsf - freq
1000
1500
2000
changerate
2500
Sampling: Composite Signal
x ( t ) = cos(2 1200t ) + cos(2 600t )
x [ n ] = cos(2 �1200 � Tsn ) + cos(2 � 600 � Tsn )
1
-1500
x ( t ) = cos(2 1200t ) + cos(2 600t )
x [ n ] = cos(2 �1200 � Tsn ) + cos(2 � 600 � Tsn )
f s = 3800Hz
fo=1200Hz, f1=600Hz
fs=2200 Hz,
1
f s = 2200Hz
fo=1200Hz, f1=600Hz
0.5
0.5
0
0
-0.5
-0.5
-1
o +freq1
x -freq1
d +freq2
+ -freq2
R -copy
B orig 2A
G +copy
0
0.2
0.4
0.6
0.8
1
t - sec
1.2
1.4
1.6
1.8
2
x 10-3
oversampling
1
0.8
0.6
0.4
0.2
o +freq1
x -freq1
d +freq2
+ -freq2
K 2nd - copy
R 1st - copy
B orig
G 1st +copy
K 2nd +copy
-1
0
0.2
0.4
0.6
0.8
1
t - sec
1.2
1.4
1.6
1.8
-3
1200Hz folded
600Hz oversample
folding
1
2
x 10
0.8
0.6
0.4
0.2
0
-1.5
-1
-0.5
0
f/fs - freq
0.5
1
1.5
0
-1.5
-1
-0.5
0
f/fs - freq
0.5
1
1.5
Resampling
Sampling: Composite Signal
x ( t ) = cos(2 1200t ) + cos(2 600t )
x [ n ] = cos(2 �1200 � Tsn ) + cos(2 � 600 � Tsn )
fs=1000 Hz,
1
fs = 1000Hz
fo=1200Hz, f1=600Hz
Resized (nearest­
Neighbor)
no pre-blur
0.5
0
-0.5
o +freq1
x -freq1
d +freq2
+ -freq2
K 2nd - copy
R 1st - copy
B orig
G 1st +copy
K 2nd +copy
-1
0
0.2
0.4
0.6
0.8
1
t - sec 1.2
1.4
1.6
1.8
2
x 10-3
folding
1
1200Hz folded
600Hz fold
0.8
Resized (nearestNeighbor)
with pre-blur
0.6
0.4
0.2
0
-1.5
-1
-0.5
0
f/fs - freq
0.5
1
Psychophysics:Vision
1.5
Courtesy of Sean T. McHugh. Used with permission.
Aliased Voice Expt.
Color Matching
Surround field
Psychophysics
Test light
the relationship between physical stimuli and what you perceive
absolute thresholds, discrimination thresholds, and scaling.
T
Bipartite
white
screen
Subject
Threshold: the point of intensity that you can just detect the presence of, or difference in, a stimulus.
absolute threshold -- level at which the subject is able to detect the presence of the stimulus (50%)
R
difference threshold -- the difference between two stimuli of differing intensities you can detect (50%)
adjust one stimulus until it is perceived as the same as the other,
describe the magnitude of the difference between two stimuli
detect a stimulus against a background.
discrimination experiments --what point the difference between two stimuli is detectable.
RGB
mixed
light
G
B
Primary lights
Figure by MIT OpenCourseWare. After Wandell, Foundations of Vision.
http://en.wikipedia.org/wiki/Psychophysics
Turn knobs for R,G, &B until combined light matches
color of test light.
http://www.ling.upenn.edu/courses/ling525/color_vision.html
Psychophysics: Vision
Psychophysics: Vision
Unsharp Mask
200
50
00
50
0
20
40
Courtesy of Wayne Fulton, http://www.scantips.com. Used with permission.
250
By exaggerating contrast along edges in the image, the edges stand out more, making them
200
appear sharper.”
50
00
50
0
0
20
40
Psychophysics: Vision
Unsharp Mask
before
Psychophysics: Vision
Unsharp Mask
200
200
50
50
original
00
50
0
20
40
00
50
+
0
20
40
after
unsharp
mask
invert
+
blur
250
200
50
00
50
0
250
“unsharp mask”
200
50
00
50
0
20
40
0
0
20
40
Psychoacoustics: Sound Masking
You know I can't hear you when the water is running!
This statement carries the essentials of the conventional wisdom about sound masking. Low-
frequency, broad banded sounds (like water running) will mask higher frequency sounds which
are softer at the listener's ear (a conversational tone from across the room)
Image removed due to copyright restrictions.
http://hyperphysics.phy-astr.gsu.edu/hbase/sound/mask.html
Chirp
+1200Hz
Broadband white noise tends to mask all frequencies, and is approximately linear in that masking.
By linear you mean that if you raise the white noise by 10 dB, you have to raise everything else
10 dB to hear it. -- http://hyperphysics.phy-astr.gsu.edu/hbase/sound/mask.html
Psychoacoustics
The following MATLAB function performs a simple
psychoacoustic test. It creates bandlimited noise, centered at 1000
Hz and also creates a sinusoid. It then plays the noise alone and
then the noise plus the sinusoid. Try different values of f and A to
see whether you can detect the sinusoid. For a particular value of
f we’ll call Amin(f) the minimum amplitude at which the
frequency f sinusoid could still be heard. Plot several values on
the graph of f vs. Amin to determine a simple masking curve.
A typical masking experiment might proceed as follows. A
short, about 400 msec, pulse of a 1,000 Hz sine wave
acts as the target, or the sound the listener is trying
to hear. Another sound, the masker, is a band of noise
centered on the frequency of the target (the masker
could also be another pure tone). The intensity of the
masker is increased until the target cannot be heard
Psychoacoustics
Minimum audible sinusoid amplitude with
noise centered at 1000Hz (A=1)
mask.m plays noise, then noise plus a sinewave of amplitude A
and frequency f. You are supposed to determine the smallest
amplitude that you can hear the sinewave over the noise. Repeat
this for several values of f, and plot f vs. Amin.
0.25
0.20
mask.m is linked in the assignments section.
0.15
Amin
Tip:
In mask.m change
sound(wf,22050)
A
%add so you get feedback of A
%pause
%comment out, so you don’t wait
sound(wf+s,22050)
>> f=1000; for A=0.01:.01:.1; mask(f,A); end
Press cmd+. to stop when you detect sinewave.
0.10
0.05
0.00
0
2000
4000
6000
f
8000
10000
12000
f
Amin
250
0.0003
500
0.00003
750
0.006
825
0.03
900
0.19
1000
0.215
1100
0.19
1250
0.08
1500
0.074
1750
0.052
2000
0.07
3000
0.04
4000
0.014
5000
0.0023
6000
0.001
7000
0.0012
8000
0.0007
9000
0.0004
10000
0.0007