ELEN E4810: Digital Signal Processing
Topic 11:
Continuous Signals
1. Sampling and Reconstruction
2. Quantization
Dan Ellis
2013-12-04
1
1. Sampling & Reconstruction

x(t)
DSP must interact with an analog world:
A to D
ADC
x[n]
DSP
y[n]
Anti- Sample
alias and
filter hold
Sensor
Dan Ellis
D to A
y(t)
DAC
Reconstruction
filter
WORLD
2013-12-04
Actuator
2
Sampling: Frequency Domain

Sampling: CT signal → DT signal by
recording values at ‘sampling instants’:
Discrete
g[n] = ga (nT )
Continuous
Sampling period T
→ samp.freq. ≠samp = 2º/T rad/sec

What is the relationship of the spectra?
≠ in rad/second

i.e. relate Ga ( j) =
and
Dan Ellis

 jt
g
t
e
 a ( ) dt

G(e ) =  g[n]e
j
CTFT
 jn
DTFT

! in rad/sample
2013-12-04
3
Sampling

DT signals have same ‘content’ as
CT signals gated by an impulse train:
ga(t)
gp(t)
×
t
t
CT delta fn
p(t)

=

n=
 (t  nT )
T
t
‘sampled’ signal:
still continuous
- but discrete
values
gp(t) = ga(t)·p(t) is a CT signal with the
same information as DT sequence g[n]
Dan Ellis
2013-12-04
4
Spectra of sampled
signals
⇥


Given CT gp (t) =
CTFT Spectrum
n= ⇥
Gp (j ) = F{gp (t)} =
ga (nT ) · (t
by linearity
ga (nT )F{ (t nT )}
n
Gp (j ) =

Compare to DTFT
ga (nT )e
⇥n
G(ej ) =
g[n]e
⇥n

i.e. G(ej ) = Gp (j )|
Dan Ellis
nT )
T=
2013-12-04
j nT
j n
≠ ~ !/T
º/T = ≠samp/2
! ~ ≠T
º
5
Spectra of sampled signals
 Also, note that
p(t ) =   (t  nT )
n
is periodic, thus has Fourier Series:
1
ck =
T
1
=
T

1
j ( 2T ) kt
p(t ) =  e
T k=

 j2 kt / T
p
t
e
dt
(
)
T / 2
T /2
shift in
But F {e j0 t x (t )} = X ( j (  0 )) frequency
(
so G p ( j) = T1 k Ga j (  k samp )
)
- scaled sum of replicas of Ga(j≠)
shifted by multiples of sampling frequency ≠samp
Dan Ellis
2013-12-04
6
CT and DT Spectra
j
1
  k 2
 So: G (e
=
G
j
=
G
j
(
)
(
(

) p T = T k a T T ))
DTFT G ( e jT ) = 1
CTFT
G
j


k
(
)
(
)

T
a
samp
k
ga (t )  Ga ( j)
−≠M
×
p(t )  P ( j)
=
−2≠samp
( )
g[n]  G e j
Dan Ellis
−2º
=
2≠samp
shifted/scaled copies
≠
2
samp
T
2013-12-04
have = T = 2
At
≠
≠samp
M/T
−4º
≠
≠M
−≠samp
g p (t )  G p ( j)
↓
ga(t) is bandlimited
@ ≠Μ
M
=
2º
4º
7
!
Avoiding aliasing

Sampled analog signal has spectrum:
Gp(j≠)
Ga(j≠)
−≠samp −≠M
−≠samp + ≠M

≠M
“alias” of “baseband”
signal
≠
≠samp
≠samp− ≠M
ga(t) is bandlimited to ± ≠M rad/sec
When sampling frequency ≠samp is large...
→ no overlap between aliases
→ can recover ga(t) from gp(t)
by low-pass filtering

Dan Ellis
2013-12-04
8
Aliasing & The Nyquist Limit

If bandlimit ≠M is too large, or sampling
rate ≠samp is too small, aliases will overlap:
Gp(j≠)
−≠M
−≠samp



≠M ≠samp
≠samp− ≠M
≠
Spectral effect cannot be filtered out
→ cannot recover ga(t)
Sampling theorum
Avoid by:  samp   M   M   samp  2 M
i.e. bandlimit ga(t) at ≤  samp Nyquist
2 frequency
Dan Ellis
2013-12-04
9
Anti-Alias Filter



To understand speech, need ~ 3.4 kHz
→ 8 kHz sampling rate (i.e. up to 4 kHz)
‘space’ for
Limit of hearing ~20 kHz
filter rolloff
→ 44.1 kHz sampling rate for CDs
Must remove energy above Nyquist with
LPF before sampling: “Anti-alias” filter
A to D
ADC
Anti-alias Sample
filter
& hold
Dan Ellis
2013-12-04
10
M
Sampling Bandpass Signals

Signal is not always in ‘baseband’
around ≠ = 0 ... may be at higher ≠:
A
−≠H −≠L

≠L ≠H
≠
Bandwidth ¢≠ = ≠H − ≠L
If aliases from sampling don’t overlap,
no aliasing distortion; can still recover:
A/T
−≠samp

≠samp/2 ≠samp
≠
2≠samp
Basic limit: ≠samp/2 ≥ bandwidth ¢≠
Dan Ellis
2013-12-04
11
Reconstruction



To turn g[n]
back to g^a(t):
g[n]
n
make a continuous
impulse train g^ p(t)
lowpass filter to
extract baseband
→ g^ a(t)

G(ej!)
º
!
G^ p(j≠)
g^
p(t)
t
^
Ga(j≠)
^
ga(t)
≠samp/2
≠
≠
t
Ideal reconstruction filter is brickwall


Dan Ellis
i.e. sinc - not realizable (especially analog!)
use something with finite transition band...
2013-12-04
12
2. Quantization

Course so far has been about
discrete-time i.e. quantization of time

Computer representation of signals also
quantizes level (e.g. 16 bit integer word)

Level quantization introduces an error
between ideal & actual signal → noise

Resolution (# bits) affects data size
→ quantization critical for compression

Dan Ellis
smallest data
coarsest quantization
2013-12-04
13
Quantization

x
Quantization is performed
in A-to-D:
x
t

A/D
n
T
Quantization has simple transfer curve:
5
Q{x} 4
3
2
1
0
-1
-2
-3
-4
-5
-5 -4 -3 -2 -1 0 1 2 3 4 5
Quantized signal
e = x - Q{x}
Quantization error
e = x  xˆ
x
Dan Ellis
xˆ = Q{x}
2013-12-04
14
Quantization noise

Can usually model quantization as
additive white noise:
i.e. uncorrelated
with self or signal x
x[n]
+
-
^
x[n]
e[n]
bits ‘cut off’ by
quantization;
hard amplitude limit
Dan Ellis
2013-12-04
15
Quantization SNR


Common measure of noise is
Signal-to-Noise ratio (SNR) in dB:
signal power
 2x
SNR = 10  log10 2 dB
e
noise power
When |x| >> 1 LSB, quantization noise
has ~ uniform distribution:
P(e[n])
2

  e2 =
12
Dan Ellis
- /
2
+ /
2
(quantizer step = ")
2013-12-04
16
Quantization SNR
Now, σx2 is limited by dynamic range of
converter (to avoid clipping)
 e.g. b+1 bit resolution (including sign)
output levels vary -2b·" .. (2b-1)"
–R
R
FS
! =
! !
where full-scale range
...! FS
b+1
2
2
RFS = 2

SNR = 10 log10
2
x
2
RFS
6b + 16.8
22(b+1) ·12
Dan Ellis
i.e. ~ 6 dB
SNR per bit
2013-12-04
20 log10
RFS
depends
on signal
17
x
Coefficient Quantization

Quantization affects not just signal
but filter constants too



.. depending on implementation
.. may have different resolution
Some coefficients
are very sensitive
to small changes

Dan Ellis
e.g. poles near
unit circle
high-Q pole
becomes
unstable
z-plane
2013-12-04
18
12/9 Project Presentations
10:15 Arthur Argall: Genomic Signal Processing
10:25 Yue Hou, Dongxue Liu:
Blind Signal Separation
10:35 Nathan Lin:
Spectral Domain Phase Microscopy
10:45 Minda Yang, Yitong Li:
Reconstruction from Neural Measurements
10:55 Yinan Wu: Mixed Channel Music
11:05 Maja Rudolph, Preston Conley:
Walking Pace Extraction from Video
11:15 Alan Zambeli-Ljepović, Tanya Shah, Sophie Wang:
Lung Sounds Analysis
Dan Ellis
2013-12-04
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