sampling, quantization -...
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Infocommunication
Sampling,
Quantization
- Bálint TÓTH, BME TMIT -
Overview
PPT is for demonstration, not for learning!
Analog signals – problem: noise, distortion
Digital signals – what are the benefits?
We’re going to talk about A/D and D/A conversions
Sampling
– The sampling theorem
– Band limited signals
Quantization
– Quantization noise
– Signal-to-noise ratio (SNR)
Bálint TÓTH – BME TMIT 2
Analog and digital signals
Sampling makes us able to use modern
technology: Audio: CD, MP3, cell phone Video: DVD
Pictures: digital camera, printer Other: Computer, etc.
Sampling
Bálint TÓTH – BME TMIT 3
Source: www.wikipedia.org
Effects of sampling (example 1)
Picture
Bálint TÓTH – BME TMIT 4
Effects of sampling (example 2)
Newspaper
Bálint TÓTH – BME TMIT 5
Effects of sampling (example 3)
Your eyes
Bálint TÓTH – BME TMIT 6
Source: eagleeye.co.ke
Effects of sampling (example 3)
Your eyes
Bálint TÓTH – BME TMIT 7
Effects of sampling (example 4)
Music
Bálint TÓTH – BME TMIT 8
Analog-to-digital conversion
Relation between the analog and the sampled
signal (Sampling theorem):
𝑋𝑆 𝑓 = 𝑓𝑠 𝑋(𝑓 − 𝑖 ∗ 𝑓𝑠)
𝑖
Bálint TÓTH – BME TMIT 9
Jean Baptiste Joseph Fourier
Bálint TÓTH – BME TMIT 10
Source: www.wikipedia.org
Who’s theorem?
The sampling theorem is usually known as the Shannon Sampling
Theorem due to Claude E. Shannon’s paper “A mathematical
theory of communciation” (1948). However, he himself said that
“…is common knowledge in the communication art.”
The minimum required sampling rate fs (i.e. 2xB) is known as the
Nyquist sampling rate or Nyquist frequency because of H.
Nyquist’s work on telegraph transmission in 1924 with K.
Küpfmüller.
The first formulation of the sampling theorem precisely and applied
it to communication is probably a Russian scientist by the name of
V. A. Kotelnikov in 1933.
However, mathematician already knew about this in a different form
and called this the interpolation formula. E. T. Whittaker published
the paper “On the functions which are represented by the
expansions of the interpolation theory” back in 1915.
Bálint TÓTH – BME TMIT 11
Digital-to-analog conversion
𝑥 𝑡 → 𝑥𝑘 → 𝑥𝑠 𝑡
The spectrum of a sampled signal:
𝑋𝑆 𝑓 = 𝑥𝑘𝑒−𝑖2𝜋𝑓𝑘𝑇
𝑘
time between samples: T
sampling frequency: fs= 1/T
Bálint TÓTH – BME TMIT 12
Band limited signal, low-pass filter
There is no spectral overlapping if
𝑋 𝑓 − 𝑖 ∗ 𝑓𝑆 = 0, 𝑖𝑓 𝑓 < −𝐵 𝑜𝑟 𝑓 > 𝐵, 𝑖 > 0
With ideal low-pass filter the reconstruction is
granted if
𝑓𝑠≥ 2𝐵
(Nyquist frequency)
Bálint TÓTH – BME TMIT
13
Sub- and oversampling
Bálint TÓTH – BME TMIT
Below the Nyquist-frequency the sampling
points are farther from each other, the signal
cannot be perfectly reconstructed (with low-
pass filter)
– Subsampling
Above the Nyquist-frequency the sampling
points are nearer to each other, the signal can
be better recognised from the samples.
– Oversampling
14
Aliasing, leakage
This phenomena is caused by the poor choise
of sampling frequency or by non-ideal filters.
Aliasing: input filter (anti-aliasing filter).
Leakage: output filter.
Bálint TÓTH – BME TMIT
15
Aliasing example
Bálint TÓTH – BME TMIT 16
Aliasing example – with antialising
Bálint TÓTH – BME TMIT 17
Aliasing example – w/o antialising
Bálint TÓTH – BME TMIT 18
Quantization
Convert the signal to discrete values
Bálint TÓTH – BME TMIT 19
Source: www.cnx.org
Quantization interval and steps
Bálint TÓTH – BME TMIT
∆ = 2m / N
20
m
Noise (linear quantization)
Power of noise:
Signal-to-noise ration for sinusoid signal:
SNR = 1.74 + 6.02n [dB]
Bálint TÓTH – BME TMIT
21
Signal transmission and reconstruction with D/A and A/D
Bálint TÓTH – BME TMIT
Filter Sampler &
holder Quantization Coder
Decoder Filter
. . .
. . . 22
Signal transmission and reconstruction with D/A and A/D
Bálint TÓTH – BME TMIT
Filter Sampler &
holder Quantization Coder
Decoder Filter
. . .
. . . 23
Signal transmission and reconstruction with D/A and A/D
Bálint TÓTH – BME TMIT
Filter Sampler &
holder Quantization Coder
Decoder Filter
. . .
. . . 24
Signal transmission and reconstruction with D/A and A/D
Bálint TÓTH – BME TMIT
3, 5, 7, 6, 4, 4, 5, 3, 2
Filter Sampler &
holder Quantization Coder
Decoder Filter
. . .
. . . 25
Signal transmission and reconstruction with D/A and A/D
Bálint TÓTH – BME TMIT
3, 5, 7, 6, 4, 4, 5, 3, 2
Filter Sampler &
holder Quantization Coder
Decoder Filter
. . .
. . .
26
Signal transmission and reconstruction with D/A and A/D
Bálint TÓTH – BME TMIT
Filter Sampler &
holder Quantization Coder
Decoder Filter
. . .
. . .
27
Signal transmission and reconstruction with D/A and A/D
Bálint TÓTH – BME TMIT 28
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