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Eeng 360 1
Illustration of the Quantization Error
Eeng 360 2
Uniform Quantization• Most ADC’s use uniform
quantizers.• The quantization levels of a
uniform quantizer are equally spaced apart.
• Uniform quantizers are optimal when the input distribution is uniform. When all values within the Dynamic Range of the quantizer are equally likely.
Input sample X
Example: Uniform =3 bit quantizer
q=8 and XQ = {1,3,5,7}
2 4 6 8
1
5
3
Output sampleXQ
-2-4-6-8
Dynamic Range:
(-8, 8)
7
-7
-3
-5
-1
Quantization Characteristic
Eeng 360 3
Quantization Example
Analogue signal
Sampling TIMING
Quantization levels. Quantized to 5-levels
Quantization levelsQuantized 10-levels
Eeng 360 4
PCM encoding example
Chart 1. Quantization and digitalization of a signal.
Signal is quantized in 11 time points & 8 quantization segments.
Chart 2. Process of restoring a signal.PCM encoded signal in binary form:101 111 110 001 010 100 111 100 011 010 101Total of 33 bits were used to encode a signal
Table: Quantization levels with belonging code words
Levels are encoded using this table
M=8
Eeng 360 5
Encoding• The output of the quantizer is one of M possible signal levels.– If we want to use a binary transmission system, then we need to map
each quantized sample into an n bit binary word.
• Encoding is the process of representing each quantized sample by an bit code word.– The mapping is one-to-one so there is no distortion introduced by
encoding.– Some mappings are better than others.
• A Gray code gives the best end-to-end performance.• The weakness of Gray codes is poor performance when the sign bit
(MSB) is received in error.
22 , log ( )nM n M
Eeng 360 6
Gray Codes
• With gray codes adjacent samples differ only in one bit position.• Example (3 bit quantization):
XQ Natural coding Gray Coding +7 111 110 +5 110 111 +3 101 101 +1 100 100 -1 011 000 -3 010 001 -5 001 011 -7 000 010
• With this gray code, a single bit error will result in an amplitude error of only 2.– Unless the MSB is in error.
Eeng 360 7
Waveforms in a PCM system for M=8
M=8
(d) PCM Signal
(c) Error Signal
(b) Analog Signal, PAM Signal, Quantized PAM Signal
(a) Quantizer Input output characteristics
22 log ( )
is the number of Quantization levels
is the number of bits per sample
nM n M
M
n
Quantization
• Analog source has infinite levels• In order to process the source output digitally, the
source have to be quantized to a finite number of levels– Reduce number of bits to a finite number– Introduce some distortion– Quantization error can be modeled as White process– The information lost in the quantization process can never
be recovered
Classification of quantization
• Scalar Quantization– Each source output is quantized individually– Further Divided into
• Uniform quantization– Equal quantization region
• Nonuniform quantization– Various length quantization region
• Vector quantization– Blocks of source output are quantized
Scalar Quantization
• ( )i i ix Q x x
1
2
3
N
1x
2x
3x
Nx
x Q(x)
Quantization Error
ix x
Signal-to-Quantization-Noise Ratio
• Mean-Square Quantization Error– – Where fx(x) denotes pdf(probability density
function) of the source random variables• SQNR(Signal-to-Quantization-Noise Ratio)
–
2
1
( ) ( )i
N
i Xi
D x x f x dx
2
10
[ ]10log
dB
E XSQNR
D
Uniform Quantization
• Equal quantization range
1 ( ~ ]a
2 ( , ]a a
3 ( , 2 ]a a
( ( 2) , )N a N
1x
2x
3x
Nx
a
Uniform Quantization
• Optimal Quantization Level– The Centroid of the interval (Center of mass)
•
• Design of the uniform quantizer is equivalent to determining a and – Then we can easily calculate SQNR
• If X has uniform distribution– Then Quantization level = midpoints of quantization range
(a + (i-2) + /2)
( )[ | ]
( )i
i
X
i i
X
xf x dxx E X X
f x dx
Uniform Quantization
• Symmetric pdf with even N (N=6)– Uniform Vs. Gaussian
a+4
a+3
a+2
a+
a
-x
=0
6x
5x
4x
3x
2x
1x
a+4
a+3
a+2
a+
a
-x
=0
6x
5x
4x
3x
2x
1x
Center is givenOnly one
parameter is chosen to
minimize distortion
Nonuniform Quantization
• Uniform Vs. Nonuniform
a+3
a+2
a+
a
-
x
4x
3x
2x
1x
5x
=0
a4
a3
a2
a1
-x
4x
3x
2x
1x
5x
=0
DifferentQuantization
Region
UniformQuantization
Region
Nonuniform Quantization
• Superior than Uniform quantization– Less Distortion– Higher SQNR
• Lloyd-Max Conditions for optimal condition–
– Choose initial quantization level:– Repeat until Minimum distortion
• Calculate quantization region boundary:• Calculate quantization level:
1
1
1( )
,2( )
i
i
i
i
a
Xa i ii ia
Xa
xf x dx x xx a
f x dx
, 1ix i N
, 1ix i N , 1ia i N
Homework
• Illustrative Problem 4.8• Problems
– 4.12
Uniform Quantization
• Symmetric pdf with even N (N=5)– Uniform Vs. Gaussian
a+3
a+2
a+
a
-
x
4x
3x
2x
1x
5x
=0
a+3
a+2
a+
a
-
x
4x
3x
2x
1x
5x
=0
Homework
• Illustrative Problem– 4.4, 4.5, 4.7
• Problems– 4.6, 4.7
20
2. Audio2.1 Human Perception2.2 Audio Bandwidth2.3 Digitization2.4 Audio Compression
2.4.1 Differential PCM2.4.2 Adaptive Differential PCM2.4.3 MP3
Contents
21
• Audio: speech, music or synthesized audio.• Audio signals are analog.• Audio Perception
Sound waves generate air pressure oscillations. Stimulate human auditory system. Transform to neural signals recognizable by the
brain.
2.1 Human Perception
22
• Features of human auditory system:1. Frequency range: Human can listen to
audio signals within the typical frequency range 20 -- 20,000 Hz.
2. Dynamic range: It is the range of the softest to the loudest audio amplitude that human can hear.
Different persons may have different frequency and dynamic ranges.
2.1 Human Perception
23
2.2 Audio Bandwidth
• Period and FrequencyA periodic signal consists of a continuously
repeated waveform pattern. If its period is T, its frequency is:
Example: The following signals are periodic with period T and frequency
Tf
1
Tf
1
24
2.2 Audio Bandwidth
25
• Signal CharacteristicA signal can be decomposed into many sinusoidal signal components such that different components1. have different frequencies and2. may have different amplitudes.
(This decomposition can be done by mathematical techniques called Fourier series and Fourier transform.)
2.2 Audio Bandwidth
26
2.2 Audio Bandwidth
Frequency of 1st component (1st harmonic) = f1 = 1/TFrequency of 2nd component (2nd harmonic)Frequency of 3rd component (3rd harmonic)= 3 f1
= 5 f1
27
2.2 Audio Bandwidth
• Frequency Domain After decomposing a signal into its components, we can analyze the
properties of this signal in the frequency domain.
Example:
It is difficult to visualize the energy content of a signal in the time domain, but it is easy to do so in the frequency domain.
28
2.2 Audio Bandwidth
• BandwidthBandwidth is the range of component frequencies.
Example:
A signal may have infinite number of components. In this case, bandwidth is defined to be the frequency range
over which x% (say, 99%) of the energy of the signal lies.
29
2.2 Audio Bandwidth
• Effect of Limited Bandwidth
If a network does not have sufficient bandwidth to send all the frequency components of a signal
some frequency components are omitted the signal is distorted.
If a network has a larger bandwidth to send more frequency components of an audio signal
the audio signal is relatively less distorted.
30
31
2.3 Digitization
• Digitization: convert an analog audio signal to digital form via sampling and quantization.SamplingSample the magnitude of the audio signal at a certain
rate.
32
2.3 Digitization
Nyquist Theorem: For a signal that has no frequency components higher than x Hz, its analog signal can be completely reproduced from its samples taken at the rate 2× of samples per second.
Illustration of Nyquist sampling rate:
33
2.3 Digitization
Example
Telephone systems transmit voice signal components with at most 4000 Hz. Sampling rate should be 8000 samples/sec.
34
• Quantization
If N bits are used to represent a sample value, there are 2N distinct quantization values.
Each sample value is rounded to the nearest quantization value, so there may be quantization error.
2.3 Digitization
35
2.3 Digitization
If the first sample value is 24.1, it is quantized to 24 (0001 1000), so the quantization error is 0.1.
36
2.3 Digitization
Pulse Code Modulation (PCM) PCM: perform sampling and quantization on audio signals. PCM is used in:
Digital telephone networks: Use a sampling rate of 8000 samples per second and 8 bits per sample, so the data rate is 64 kbps (adopted in ITU-T G.711).
Audio CD: Use a sampling rate of 44100 samples per second and 16 bits per sample, so the data rate for stereo audio is 1.411 Mbps.
37
2.4 Audio Compression
2.4.1 Differential PCM
Differential PCM is a compressed version of PCM. It haslower bit rate but its voice quality may be poorer.
Differential PCMVoice signal changes slowly compared with the sampling rate.
Successive sample values have a small difference.
Use fewer bits to encode the difference between the current sample value and the previous one.
Lower bit rate, but voice quality may be degraded when voice amplitude changes abruptly.
38
2.4 Audio Compression
Example
For PCM in digital telephony, sampling rate is 8000 samples/sec and 8 bits are used for each sample. Data rate is 64 kbps.
If differential PCM is adopted and 6 bits are used to encode the difference between successive sample values, data rate is reduced to 48 kbps.
39
2.4 Audio Compression
2.4.2 Adaptive Differential PCM
Adaptive differential PCM is an improved version of differential PCM.
Main idea: When the voice amplitude changes steeply for a significant duration, change to use a larger quantization step (i.e., a larger difference between successive quantization values)
40
2.4 Audio Compression
41
2.4 Audio Compression
ITU-T G.721 adopts adaptive differential PCM, a sampling rate of 8000 samples per second, and 4 bits for encoding thedifference between successive sample values.
Bit rate is 32 kbps, but voice quality is only slightly worse than that in PCM at 64 kbps.
42
2.4 Audio Compression
2.4.3 MP3
CD audio has a data rate of 1.411 Mbps. Well-known compression method for CD audio: MP3.
MP3: MPEG audio layer 3. (MPEG specifies three audio compression layers.)
MP3 adopts perceptual coding to attain a high compression ratio and provide very good audio quality.
43
2.4 Audio Compression
Perceptual Coding
It is based on the science of psychoacoustics, which studies how people perceive sound.
It exploits certain flaws in the human auditory system for compression, such that the compressed audio sounds about the same to human even though its signal waveform may become quite different.
44
2.4 Audio Compression1st Flaw: Threshold of Audibility
When a frequency component is very weak (i.e., its power is below a threshold), human cannot hear it.
Threshold of audibility (averaged over many people)
Compression: Omit the frequency components whose power falls below the threshold of audibility.
45
2.4 Audio Compression2nd Flaw: Frequency Masking
Some sounds can mask other sounds: a loud sound in one frequency band hides a softer sound in another frequency band.
Masking effect:
Compression: Omit the masked frequency components.
46
2.4 Audio Compression3rd Flaw: Temporal Masking
When a masking sound ends, it takes a short time before hearing the masked sound.
Masking effect:
Compression: If the amplitudes of the masked frequency components are less than the decay envelope, omit these components.
47
2.4 Audio Compression
To use MP3 for compression, we select two options:
Sampling rate: We can sample the waveform at 32 kHz, 44.1 kHz or 48 kHz on one or two channels.
Bit rate: Typically, we choose the bit rate to be 96 kbps, 128 kbps or 160 kbps.
48
2.4 Audio Compression
Main Steps for Compression
Perform sampling on the audio signal. Divide the samples into groups with 1152 samples per group.
Each group is passed through: (i) 32 digital filters to get 32 frequency subbands, and (ii) a psychoacoustic model to determine the masked frequencies.
Based on the available "bit budget" (depending on the chosen bit rate), allocate more bits to the subbands with larger unmasked spectral power.
Finally, use Huffman coding to encode the bits (i.e., assign shorter codewords to numbers that appear frequently).
Example
Networks: Data Encoding 50
Networks: Data Encoding 51
PCMNonlinear Quantization Levels
Networks: Data Encoding 52
Delta Modulation DCC 6th Ed. W.Stallings
Analog Representations of SoundMagnified phonograph grooves, viewed from above:
The shape of the grooves encodes the continuously varying audio signal.
Analog to Digital Recording Chain
ADC
Continuously varying electrical energy is an analog of the sound pressure wave.
Microphone converts acoustic to electrical energy. It’s a transducer.
ADC (Analog to Digital Converter) converts analog to digital electrical signal.Digital signal transmits binary numbers.
DAC (Digital to Analog Converter) converts digital signal in computer to analog for your headphones.
Analog versus Digital
AnalogContinuous signal that mimics shape of acoustic sound pressure wave
DigitalStream of discrete numbers that represent instantaneous amplitudes of analog signal, measured at equally spaced points in time.
Analog to Digital Conversion
Instantaneous amplitudes of continuous analog signal, measured at equally spaced points in time.
A series of “snapshots”
[a.k.a. “sample word length,” “bit depth”]Precision of numbers used for measurement: the more bits, the higher the resolution.
Example: 16 bit
Analog to Digital Overview
Sampling RateHow often analog signal is measured
Sampling Resolution
[samples per second, Hz]Example: 44,100 Hz
Sampling Rate
Nyquist Theorem:
Sampling rate must be at least twice as high as the highest frequency you want to represent.
Determines the highest frequency that you can represent with a digital signal.
Capturing just the crest and trough of a sine wave will represent the wave exactly.
Aliasing
What happens if sampling rate not high enough?
A high frequency signal
sampled at too low a rate
looks like …
… a lower frequency signal.
That’s called aliasing or foldover. An ADC has a low-pass anti-aliasing filter to prevent this.
Synthesis software can cause aliasing.
Common Sampling Rates
Sampling Rate Uses
44.1 kHz (44100) CD, DAT
48 kHz (48000) DAT, DV, DVD-Video
96 kHz (96000) DVD-Audio
22.05 kHz (22050) Old samplers
Most software can handle all these rates.
Which rates can represent the range of frequencies audible by (fresh) ears?
4-bit QuantizationA 4-bit binary number has 24 = 16 values.
0
2
4
6
8
10
12
14
Ampl
itude
A better approximation
Time — measure amp. at each tick of sample clock
Quantization Noise
Round-off error: difference between actual signal and quantization to integer values…
Random errors: sounds like low-amplitude noise
The Digital Audio Stream
It’s just a series of sample numbers, to be interpreted as instantaneous amplitudes: one for every tick of the sample clock.
Previous example:11 13 15 13 10 9 6 1 4 9 15 11 13 9
This is what appears in a sound file, along with a header that indicates the sampling rate, bit depth and other things.
Common Sampling Resolutions
Word length Uses
8-bit integer Low-res web audio
16-bit integer CD, DAT, DV, sound files
24-bit integer DVD-Video, DVD-Audio
32-bit floating point Software (usually only for internal representation)
16-bit Sample Word Length
A 16-bit integer can represent 216, or 65,536, values (amplitude points).
We typically use signed 16-bit integers, and center the 65,536 values around 0.
32,767
-32,768
0
Audio File Size
CD characteristics…- Sampling rate:
44,100 samples per second (44.1 kHz)
How big is a 5-minute CD-quality sound file?
- Sample word length:16 bits (i.e., 2 bytes) per sample
- Number of channels:2 (stereo)
Audio File Size
5 minutes * 60 seconds per minute= 300 seconds
How big is a 5-minute CD-quality sound file?
44,100 samples * 2 bytes per sample * 2 channels= 176,400 bytes per second
300 seconds * 176,400 bytes per second= 52,920,000 bytes = c. 50.5 megabytes (MB)
DAC: Sample and HoldTo reconstruct analog signal, hold each sample value for one clock tick; convert it to steady voltage.
0
1
2
3
4
5
6
7
Ampl
itude
Time
DAC: Smoothing FilterApply an analog low-pass filter to the output of the sample-and-hold unit: averages “stair steps” into a smooth curve.
0
1
2
3
4
5
6
7
Ampl
itude
Time
Prof. Brian L. Evans
Dept. of Electrical and Computer Engineering
The University of Texas at Austin
EE445S Real-Time Digital Signal Processing Lab Spring 2011
Lecture 8
Quantization
8 - 71
Outline
• Introduction
• Uniform amplitude quantization
• Audio
• Quantization error (noise) analysis
• Noise immunity in communication systems
• Conclusion
• Digital vs. analog audio (optional)
8 - 72
Resolution• Human eyes
Sample received light on 2-D grid Photoreceptor density in retina
falls off exponentially awayfrom fovea (point of focus)
Respond logarithmically tointensity (amplitude) of light
• Human earsRespond to frequencies in 20 Hz to 20 kHz rangeRespond logarithmically in both intensity (amplitude)
of sound (pressure waves) and frequency (octaves)Log-log plot for hearing response vs. frequency
Foveated grid:point of focus in middle
8 - 73
Data Conversion
• Analog-to-DigitalConversionLowpass filter has
stopband frequencyless than ½ fs
• Digital-to-AnalogConversionLowpass filter has stopband
frequency less than ½ fs
Discrete-to-continuousconversion could be assimple as sample and hold
Analog Lowpass Filter
Discrete to Continuous Conversion
fs
Lecture 7
Analog Lowpass Filter
Quantizer
Sampler at sampling rate of fs
Lecture 8Lecture 4
8 - 74
Types of Quantizers• Quantization is an interpretation of a
continuous quantity by a finite set of discrete values
• Amplitude quantization approximates its input by a discrete amplitude taken from finite set of values
System Property Amplitude Quantizer
Sampler Sampler + Quantizer
Linearity Yes
Time-invariance No
Causality Yes
Memoryless Yes
For the sampler, stay in the continuous time domain at the input and output to decide on time invariance
8 - 75
Public Switched Telephone Network
• Sample voice signals at 8000 samples/s• Quantize voice to 8 bits/sample
Uniformly quantize to 8 bits/sample, orCompand by uniformly quantizing to 12 bits and
map12 bits logarithmically to 8 bits (by lookup table) to allocate more bits in quiet segments (where ear is more sensitive)
)1log(
)1log(
xy
m = 256 in US/Japan and A = 87.6 in Europe
11
log1
log1
10
log1
xA
ifA
xAA
xifA
xA
y
Maximum data rate?
kbps
m lawx1
1
y
A lawx1
1
8 - 76
Uniform Quantization• Round to nearest integer (midtread)
Quantize amplitude to levels {-2, -1, 0, 1}Step size D for linear region of operationRepresent levels by {00, 01, 10, 11} or
{10, 11, 00, 01} …Latter is two's complement representation
• Rounding with offset (midrise)Quantize to levels {-3/2, -1/2, 1/2, 3/2}Represent levels by {11, 10, 00, 01} …Step size 1
3
3
1223
23
2
13
3
12
)2(12
x
Q[x]
1-2
-2
1
x
Q[x]
1-2 -1
1
2Used in slide 8-10
8 - 77
Handling Overflow• Example: Consider set of integers {-2, -1, 0, 1}
Represented in two's complement system {10, 11, 00, 01}.
Add (–1) + (–1) + (–1) + 1 + 1Intermediate computations are – 2, 1, –2, –1 for
wraparound arithmetic and –2, –2, –1, 0 for saturation arithmetic
• Saturation: When to use it?If input value greater than maximum,
set it to maximum; if less than minimum, set it to minimum
Used in quantizers, filtering, other signal processing operators
• Wraparound: When to use it?Addition performed modulo set of integersUsed in address calculations, array indexing
Native support in MMX and DSPs
Standard two’s complement behavior
8 - 78
Audio Compact Discs (CDs)• Sampled at 44.1 kHz
Analog signal bandwidth from 0 Hz of 20 kHzAnalog signal bandwidth from 20 kHz to 22.05 kHz is
for anti-aliasing filter to roll off from passband to stopband(rolloff is about 10% of maximum passband frequency)
• Amplitude is uniformly quantized to B = 16 bits to yield signal-to-noise ratio of
1.76 dB + 6.02 dB/bit * B = 98.08 dBThis loose upper bound is derived later in slides 8-11
to 8-15In practice, audio CDs have dynamic range of about
95 dB
8 - 79
Dynamic Range• Signal-to-noise ratio in dB
• For linear systems, dynamicrange is equal to SNR
• Linear time-invariant filter for bandlimited signalPass signal bandwidth: magnitude response of 1
means 0 dBAttenuate out-of-band noise: Astopband = dynamic
range
Power Noiselog 10
Power Signallog 10 Power Noise
Power Signallog 10SNR
10
10
10dB
Why 10 log10 ?
For amplitude A,AdB = 20 log10 A
With power P A2 ,PdB = 10 log10 A2
PdB = 20 log10 A
8 - 80
Dynamic Range in Audio• Sound Pressure Level (SPL)
Reference in dB SPL is 20 Pa(threshold of hearing)
40 dB SPL noise in typical living room120 dB SPL threshold of pain 80 dB SPL resulting dynamic rangeAudio CDs give 95 dB of dynamic range
• Estimating dynamic range(a)Find maximum RMS output of the linear system
with some specified amount of distortion, typically 1%
(b)Find RMS output of system with small input signal (e.g.-60 dB of full scale) with input signal removed from output
(c)Divide (b) into (a) to find the dynamic range
Anechoic room 10 dBWhisper 30 dB Rainfall 50 dB
Dishwasher 60 dB City Traffic 85 dB
Leaf Blower 110 dB Siren 120 dB
Slide by Dr. Thomas D. Kite, Audio Precision
8 - 81
Quantization Error (Noise) Analysis• Quantization output
Input signal plus noiseNoise is difference of
output and input signals
• Signal-to-noise ratio (SNR) derivationQuantize to B bits
Quantization error
• Assumptionsm (-mmax, mmax)
Uniform midrise quantizerInput does not overload quantizerQuantization error (noise) is uniformly distributedNumber of quantization levels L = 2B is large enough
so thatQB[ · ]m v
mvmmQq B ][LL
1
1
1
8 - 82
Quantization Error (Noise) Analysis• Deterministic signal x(t)
w/ Fourier transform X(f)Power spectrum is square of
absolute value of magnitude response (phase is ignored)
Multiplication in Fourier domain is convolution in time domain
Conjugation in Fourier domain is reversal & conjugation in time
• Autocorrelation of x(t)
Maximum value (when it exists) is at Rx(0)
Rx(t) is even symmetric,i.e. Rx(t) = Rx(-t)
)( )()()( *2fXfXfXfPx
)(*)( )( )( ** xxFfXfX
)(*)()( * xxRx
t
1x(t)
0 Ts
t
Rx(t)
-Ts Ts
Ts
8 - 83
Quantization Error (Noise) Analysis• Two-sided random signal n(t)
Fourier transform may not exist, but power spectrum exists
For zero-mean Gaussian random process n(t) with variance s2
• Estimate noise powerspectrum in Matlab
22* )( )( )( )( )( fPtntnER nn
)( )( nn RFfP
N = 16384; % finite no. of samplesgaussianNoise = randn(N,1);plot( abs(fft(gaussianNoise)) .^ 2 );
approximate noise floor
dttntntntnERn )( )( )( )( )( **
)(*)( )( )( )( )( )( ***
nndttntntntnERn
8 - 84
Quantization Error (Noise) Analysis• Quantizer step size
• Quantization error
q is sample of zero-mean random process Q
q is uniformly distributed
• Input power: Paverage,m
SNR exponential in BAdding 1 bit increases
SNR by factor of 4• Derivation of SNR in
deciBels on next slide
L
m
L
m maxmax 2
1
2
22
q
BQ
zero
m
QE
22max
22
222
2 3
1
12
B
Q m
PP 22max
maverage,2
maverage, 2 3
SNR
PowerNoise
PowerSignalSNR
8 - 85
Quantization Error (Noise) Analysis• SNR in dB = constant + 6.02 dB/bit * B
• What is maximum number of bits of resolution forLandline telephone speech signal of SNR of 35 dBAudio CD signal with SNR of 95 dB
BmP
mP
m
P B
02.6log 20log 10477.0
)2(logB 20log 20log 103log 10
2 3
log 10SNR log 10
max10maverage,10
10max10maverage,1010
22max
maverage,1010
1.76 and 1.17 are common constants used in audio
Loose upper bound
8 - 86
Noise Immunity at Receiver Output• Depends on modulation, average transmit
power, transmission bandwidth and channel noise
• Analog communications (receiver output SNR)“When the carrier to noise ratio is high, an increase
in the transmission bandwidth BT provides a corresponding quadratic increase in the output signal-to-noise ratio or figure of merit of the [wideband] FM system.” – Simon Haykin, Communication Systems, 4th ed., p. 147.
• Digital communications (receiver symbol error)“For code division multiple access (CDMA) spread
spectrum communications, probability of symbol error decreases exponentially with transmission bandwidth BT” – Andrew Viterbi, CDMA: Principles of Spread Spectrum Communications, 1995, pp. 34-36.
8 - 87
Conclusion• Amplitude quantization approximates its input
by a discrete amplitude taken from finite set of values
• Loose upper bound in signal-to-noise ratio of a uniform amplitude quantizer with output of B bitsBest case: 6 dB of SNR gained for each bit added to
quantizerKey limitation: assumes large number of levels L = 2B
• Best case improvement in noise immunity for communication systemsAnalog: improvement quadratic in transmission
bandwidthDigital: improvement exponential in transmission
bandwidth
8 - 88
Digital vs. Analog Audio• An audio engineer claims to notice differences
between analog vinyl master recording and the remixed CD version. Is this possible?When digitizing an analog recording, the maximum
voltage level for the quantizer is the maximum volume in the track
Samples are uniformly quantized (to 216 levels in this case although early CDs circa 1982 were recorded at 14 bits)
Problem on a track with both loud and quiet portions, which occurs often in classical pieces
When track is quiet, relative error in quantizing samples grows
Contrast this with analog media such as vinyl which responds linearly to quiet portions
Optional
8 - 89
Digital vs. Analog Audio• Analog and digital media response to voltage v
• For a large dynamic rangeAnalog media: records voltages above V0 with
distortionDigital media: clips voltages above V0 to V0
• Audio CDs use delta-sigma modulationEffective dynamic range of 19 bits for lower
frequencies but lower than 16 bits for higher frequencies
Human hearing is more sensitive at lower frequencies
03/1
00
00
03/1
00
for
for
for
)(
VvvVV
VvVv
VvVvV
vA
00
00
00
for
for
for
)(
VvV
VvVv
VvV
vD
Optional