quantization prof. siripong potisuk. a block diagram of a dsp system
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Quantization
Prof. Siripong Potisuk
A Block Diagram of a DSP System
Analog-to-Digital Conversion (ADC)
Discretize the independent variable or time of an analog signal (Sampling)
Discretize the dependent variable or amplitude of an analog signal by rounding off to the nearest integer (Quantization)
Each quantization level represented using binary encoding scheme (Encoding)
Flash, Successive approximation, Sigma-delta
A Typical ADC Process
The process of converting analogvoltage with infinite precision to finiteprecision is called the quantization process.
Analog-to-Digital Conversion (ADC)
Quantizer Input-output Characteristics
Similar to passing a discrete-time signal through a piecewise constant staircase type function
2 types: mid-tread and mid-rise
Quantization Level• Suppose the value of x[n] ranges over the interval
[xmin, xmax]. The spacing between adjacent quantization level or step size (ADC resolution) is
• L = # of quantization levels• N = # of binary bits used to represent the value of x[n]• The resulting quantization level, xq , is
• i is an index corresponding to the binary code
N
xx
L
xx
2minmaxminmax
1,,1,0,min Liixxq
minxxroundi
Example
A speech signal has a maximum frequency of 4 kHz. We want to digitize it and send it in a file using 2 bytes (i.e., 16 bits) per sample. What would be the minimum length of the file occupied by the signal for each minute of recording? Assume the signal is not compressed.
Quantization Error• Also known as quantization noise
• Modeled as a random variable uniformly distributed over the interval with probability density p(eq) = 1/
• The average power of the quantization noise is
xxe qq
Example 2.9 & 2.10 (text)
Assuming that a 3-bit ADC accepts an analog input ranging from 0 to 5 volts, determine(a) the number of quantization levels;(b) the step size or resolution of the quantizer;(c) the quantization level corresponding to the analog value of 3.2 volts;(d) the binary code produced by the encoder.(e) the quantization error corresponding to the 3.2-V analog input.
Problem 2.27 (text)
Assuming that a 3-bit ADC accepts an analog input ranging from -2.5 to 2.5 volts, determine(a) the number of quantization levels;(b) the step size or resolution of the quantizer;(c) the quantization level corresponding to the analog value of -1.2 volts;(d) the binary code produced by the encoder.(e) the quantization error corresponding to the analog input.
Example
Suppose the amplitude of a discrete-time signal x[n] is constrained to lie in the interval [-10, 10]. If the average power of the quantization noise is to be less than 0.001, what is the minimum number of bits that are needed to represent the value of x[n]?
Signal-to-quantization Noise Ratio (SNRq)
A figure of merit expressed in terms of the ratio between signal power and the quantization noise power
Usually expressed in decibels (dB)
1
0
2
1
0
2
2
2
][1
][1
)(
)(N
nq
N
n
neN
nxN
eE
xESNR
rms
dBq
xSNRSNR 1010 log2079.10)(log10
Special Case: The signal has a full-scale dynamic range
maxminmax ||2 xxx
Thus,
max1010 ||2
2log2079.10log2079.10
x
xxSNR rms
Nrms
dBq
2log202log20||
log2079.10 1010max
10
N
x
xrms
dBNx
xrms 02.6||
log2077.4max
10
Increasing 1 bit of the ADC quantizer can improve SNRq by 6 dB The 6-dB rule
For a full-scale sinusoidal signal with amplitude A,
NNA
ASNR
dBq 02.676.102.6707.0
log2077.4 10
Example
In a DSP system, the output SNRq is to be held to a
minimum of 40 dB. Determine the number of required quantization levels, and the corresponding
SNRq assuming a full-scale sinusoidal input.
Flash ADC Unit
One of several ways to implement ADC Consists of a series of reference voltages created
by equal resistors A set of comparators is used to compare the input
voltage with the reference voltages An encoding logic unit outputs the binary sequence Offers high conversion speed Impractical for high-resolution applications
An Example of a Simple 2-bit Flash ADC
Non-uniform Quantization
Needed for signals whose smaller amplitudes predominate and larger ones are rare (i.e.,
speech) Difficult to design
Companding
Pre- and post-processing applied to a uniform quantizer to achieve non-uniform quantization
The combination of compression and expansion The signal samples first compressed before
passing through a uniform quantizer To restore the signal samples to their correct
relative level, an expander with a characteristic complementary to that of the compressor is used in the receiver
Characteristics of a compressor
Compression Law
-law (used in digital telephony in North America & Japan with = 255)
A-law (used in Europe with A = 87.6)
1),sgn()1ln(
1ln
max
max
x
xx
x
x
y
11
1
),sgn(ln1
ln1
,ln1
max
max
max
max
x
x
A
Ax
x
xA
x
xA
x
x
A
A
y