1. ELEN E4810: Digital Signal Processing Topic 11: Continuous Signals
Sampling and Reconstruction
Quantization
Dan Ellis

2. 1. Sampling & Reconstruction
DSP must interact with an analog world:
A to D
D to A
x(t)
x[n]
y[n]
y(t)
DSP
ADC
DAC
Anti-
alias
filter
Sample
and
hold
Reconstruction
filter
WORLD
Sensor
Actuator
Dan Ellis

3. Sampling: Frequency Domain
Sampling: CT signal  DT signal by recording values at ‘sampling instants’:
What is the relationship of the spectra?
i.e. relate and
Discrete
Continuous
Sampling period T
 samp.freq. WT = 2p/T rad/sec
W in rad/second
CTFT
DTFT
w in rad/sample
Dan Ellis

4. ‘sampled’ signal: still continuous
Sampling
DT signals have same ‘content’ as CT signals gated by an impulse train:
gp(t) = ga(t)·p(t) is a CT signal with the same information as DT sequence g[n]
gp(t)
ga(t)  t t
‘sampled’ signal: still continuous
- but discrete
values t
CT delta fn
p(t) T 3T
Dan Ellis

5. Spectra of sampled signals
Given CT
Spectrum
Compare to DTFT
i.e.
by linearity W
p/T = WT/2 w p
Dan Ellis

6. Spectra of sampled signals
Also, note that is periodic, thus has Fourier Series:
But so
shift in
frequency
- scaled sum of shifted replicas of Ga(jW) by multiples of sampling frequency WT
Dan Ellis

7. shifted/scaled copies
CT and DT Spectra
 So: or
DTFT
CTFT
ga(t) is bandlimited
@ WM M W
-WM WM  W
-2WT
-WT WT
2WT =
shifted/scaled copies
M/T W  w
-4p
-2p 2p 4p
Dan Ellis

8. Aliasing Sampled analog signal has spectrum: Gp(jW)
ga(t) is bandlimited to ± WM rad/sec
When sampling frequency WT is large...
 no overlap between aliases
 can recover ga(t) from gp(t) by low-pass filtering
Ga(jW)
“alias” of “baseband”
signal
Gp(jW) W
-WT
-WM WM WT
WT - WM
-WT + WM
Dan Ellis

9. The Nyquist Limit
If bandlimit WM is too large, or sampling rate WT is too small, aliases will overlap:
Spectral effect cannot be filtered out  cannot recover ga(t)
Avoid by:
i.e. bandlimit ga(t) at ≤
Gp(jW) W
-WT
-WM WM WT
WT - WM
Sampling theorum
Nyquist frequency
Dan Ellis

10. ‘space’ for filter rolloff
Anti-Alias Filter
To understand speech, need ~ 3.4 kHz
 8 kHz sampling rate (i.e. up to 4 kHz)
Limit of hearing ~20 kHz
 44.1 kHz sampling rate for CDs
Must remove energy above Nyquist with LPF before sampling: “Anti-alias” filter
‘space’ for filter rolloff
A to D
ADC
Anti-alias
filter
Sample
& hold M
Dan Ellis

11. Sampling Bandpass Signals
Signal is not always in ‘baseband’ around W = may be some higher W:
If aliases from sampling don’t overlap, no aliasing distortion, can still recover
Basic limit: WT/2 ≥ bandwidth DW M W
-WH
-WL WL WH
Bandwidth DW = WH - WL
M/T W
-WT
WT/2 WT
2WT
Dan Ellis

12. Reconstruction To turn g[n] back to ga(t):
make a continuous impulse train gp(t)
lowpass filter to extract baseband  ga(t)
Ga(ejw)
To turn g[n] back to ga(t):
Ideal reconstruction filter is brickwall
i.e. sinc - not realizable (especially analog!)
use something with finite transition band...
g[n] ^ n w p ^
gp(t) ^
Gp(jW) ^ t W
WT/2 ^
Ga(jW)
ga(t) ^ ^ t W
Dan Ellis

13. 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
smallest data  coarsest quantization
Dan Ellis

14. Quantization Quantization is performed in A-to-D:
Quantization has simple transfer curve:
Quantized signal
Quantization error
Dan Ellis

15. i.e. uncorrelated with self or signal x
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

16. Quantization SNR
Common measure of noise is Signal-to-Noise ratio (SNR) in dB:
When |x| >> 1 LSB, quantization noise has ~ uniform distribution:
signal power
noise power
(quantizer step = e)
Dan Ellis

17. Quantization SNR
Now, sx2 is limited by dynamic range of converter (to avoid clipping)
e.g. b+1 bit resolution (including sign) output levels vary -2b·e .. (2b-1)e
where full-scale range
depends
on signal
i.e. ~ 6 dB SNR per bit
Dan Ellis

18. 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
e.g. poles near unit circle
high-Q pole
becomes
unstable
Dan Ellis