digital representation of analogue signals analogue to ...€¦ · analogue to digital conversion...
TRANSCRIPT
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Digital Representation of Analogue Signals
Analogue to Digital Conversion
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Analogue to Digital Conversion
Digital signal is superior to an analog signal (why?)
Need to change an analog signal to digital data
Two common techniques
Pulse Code Modulation
Delta Modulation.
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PCM encoder
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Sampling
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Also known as
Nyquist–Shannon sampling theorem
• establishes a sufficient condition for a sample
rate that permits a discrete sequence of
samples to capture all the information from a continuous-time signal of finite bandwidth
Nyquist theorem
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Generally (what we assume in this course)
“the sampling rate must be at least 2 times the highest frequency contained in the signal”
Sometimes
“the sampling rate must be at least 2 times the width of the non-zero frequency interval” (as opposed to its highest frequency component)
More complex to implement this constraint
Very important for high-frequency signals with narrow bandwidth (e.g., one radio transmission channel)
Nyquist theorem: Details
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Nyquist theorem: Example
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Nyquist theorem: Intuition
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Nyquist theorem: Clocks
Note: We only care about ONE dimension!
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Telephone companies digitize voice by assuming a
maximum frequency of 4000 Hz. The sampling rate therefore is 8000 samples per second.
Nyquist theorem: Telephones
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What is its spectrum?
Can we sample it?
Nyquist theorem: Square Wave
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A complex bandpass signal has a bandwidth of 200
kHz. What is the minimum sampling rate for this signal?
Solution
We cannot find the minimum sampling rate in this case
because we do not know where the bandwidth starts or
ends. We do not know the maximum frequency in the
signal.
Book Example 4.11
Not True!
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PCM: Quantization and Encoding
-0.28
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PCM: Quantization and Encoding
PAM: infinite precision
PCM: precision limited by the precision of the numbers chosen for amplitude “measurement”
This imprecision results in quantization noise
Unlike many other kinds of noise, q.n. is correlated with the signal that was digitized
This correlation is bad => often need to do noise
shaping (basically, add good white or non-white noise to mask the bad quantization noise)
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PCM: Quantization and Encoding
Where does 1.76 dB come from?
Applies only to sinusoidal signals. With sinusoids,
the distribution of signal values is non uniform, and
the amount of noise generated is lower than with
uniform noise (would have been + 0 dB with random signals)
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Quantization Noise: Examples (wiki)
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Dithering in Audio in Detail
500 Hz, no dither; here and further, ~3 bits
With Dither, ±1 LSB, triangular (rand()+rand()) PDF
More noise, but it’s uniform and less obtrusive now
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Noise Shaping (2 ways)
Filter dither before adding
Move noise where it’s hard to hear
FIR filter: error @1 sample before as feedback
Quantize this, instead of x[n]:
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Is Dithering Always Needed?
Don’t care about the nature of noise – noe.g., the result is not an image or a sound
With sound, if the dynamic range is high, during the quietest passages, the highest bits stay zero –meaning, temporarily, for those parts you might really use only 2 or 3 bits, not 16!
Using dithering and noise shaping increases the dynamic range past “6.02 dB per bit” for some frequencies
When the source input is at least slightly noisy (e.g., microphone, cheap audio device), no additional dithering noise is needed
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What is the SNRdB in the example of Figure 4.26?
Solution
We can use the formula to find the quantization. We have
eight levels and 3 bits per sample, so
SNRdB = 6.02(3) + 1.76 = 19.82 dB
Increasing the number of levels increases the SNR.
Book Example 4.12
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A telephone subscriber line must have an SNRdB
above 40. What is the minimum number of bits per sample?
Solution
We can calculate the number of bits as
Example 4.13
Telephone companies usually assign 7 or 8 bits per sample.
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We want to digitize the human voice. What is the bit rate, assuming 8 bits per sample?
Solution
The human voice normally contains frequencies from 0 to
4000 Hz. So the sampling rate and bit rate are calculated
as follows:
Example 4.14
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Delta ModulationUse a reference (staircase)
Compare the signal against the [delayed] reference (1/0) and update it (up/down)
(+) simple to implement
(–) high “sampling” rate needed; only one bit: hard to process
Staircase “remembers” (integrates) the obtained bits
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Delta modulation components
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Delta demodulation components
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Similar ADC: Delta-Sigma
Quantize the signal and the accumulated error
“Remember” how far away from the true values we are and correct for this systematic “straying”
Sometimes known as PD[ensity]M
Simple to implement
Works best at (very) high sampling frequencies
Dithering and noise shaping is usually an integral part!
SACD implementation
1 bit
2.8224 MHz (64x CD’s 44.1 kHz)
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Which ADC to Use
Want to process (e.g., mixing, noise reduction, compression): bit stream output hard to use
PCM modulation
For high speed (e.g., video)
Direct Conversion (2n comparators, 8–10 bit max)
For high quality sound need high SNR
Delta-Sigma, followed by decimation into 16/24 bits
Slow and simple (sensors, measuring tools)
Ramp Compare, Integrating ADC (counter-based)
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Summary: Digitization of Analog Signals
1. Sampling: obtain samples of x(t) at uniformly spaced time intervals
2. Quantization: map each sample into an approximation value of finite precision
� Pulse Code Modulation: telephone speech
� CD audio
3. Compression: to lower bit rate further, apply additional compression method
� Differential coding: cellular telephone speech
� Subband coding: MP3 audio
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Samplert
x(t)
t
x(nT)
Interpolationfilter
t
x(t)
t
x(nT)
(a)
(b)
Nyquist: Perfect reconstruction if sampling rate 1/T > 2Ws
Reconstruction
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Digital Transmission of Analog Information
Interpolationfilter
Displayor
playout
2W samples / s
2W m bits/sx(t)
Bandwidth W
Sampling(A/D)
QuantizationAnalogsource
2W samples / s m bits / sample
Pulse
generator
y(t)
Original
Approximation
Transmission
or storage