Digital Multiplexing

1
Digital Multiplexing
2
Multiplexing

• Voice frequency conversations are multiplexed in
  traditional telephone transmission systems via
• Frequency Division Multiplexing (FDM). -
  Historical system
• Time Division Multiplexing (TDM), combined with
  digital coding of the voice signals. -Now the
  dominant method)
• Analog FDM is very similar to radio broadcasting
  (although via wires and not via an antenna)
• FDM is used in almost all radio systems,
  sometimes combined with TDMA, CDMA, etc.
• Only radio exception to FDMA is pulse position
  modulation (PPM) also called ultra wide band,,
  a recent development for radio purposes
• Each voice signal is instantaneously multiplied
  by a distinct frequency sine wave (amplitude
  modulation) for analog FDM
• Typically 12 distinct modulated carrier
  frequencies are added and transmitted via the
  same wires in telephone FDM systems
• Each voice signal has a pre-designated frequency
  receiver
• FDM was the only telephone multiplexing
  technology in use until 1961

3
FDM Analog Telephone Carrier

• Feasible with hardware available in 1920s.
• Derived from radio communication techniques
• Digital technology, although understood
  theoretically in 1930s, was not economically
  feasible until transistors and integrated
  circuits were developed as well
• FDM reached a high state of technical refinement
• Single Side Band (SSB) analog amplitude
  modulation (AM) (invented and analyzed by J.R.
  Carson) is the most spectrum-efficient method of
  modulation, using only about 4 kHz bandwidth per
  telephone voice channel
• SSB still used extensively in military and
  amateur radio
• Second and third order FDM systems hierarchy were
  then used to multiplex the lower order multiplex
  groups for more channels per link. Microwave and
  co-ax cable used FDM extensively in this way.
• FDM installations declined after 1960s, replaced
  by digital multiplexing

4
FDM Disadvantages

• Basic limitations of analog amplifier noise and
  distortion were still present
• Longer transmission distance requires more
  amplifiers. More amplifiers produces more audio
  noise and distortion
• Negative Feedback design, based on the invention
  of H.S.Black, produces a low distortion, low
  noise, (high fidelity) amplifier. The noise and
  distortion is lower but not zero!
• Although highly refined in design, hardware was
  still relatively costly to make, install, adjust
• With todays integrated circuit technology, it
  could be improved further, (examine a cellular
  radio unit, for example, which uses similar
  analog RF technology), but would not be quite as
  compact and low cost as equivalent functionality
  using digital multiplexing

5
Digital Multiplexing

• T-1, developed in 1961, quickly displaced FDM
• An almost ideal new product
• Better speech quality than analog FDM
• 24 channels double the capacity of predecessor
  12 channel analog N-carrier
• Direct replacement for 2 N-carrier links
• Installed cost about equal to N-carrier (thus
  half the cost per channel)
• Cost has since reduced even further due to use of
  integrated circuits, etc.
• Little or no field adjustment, calibration, etc.
  (low maintenance costs)
• Most new products are not simultaneously better
  in price and quality, or may have backward
  compatibility problems

6
Advantages of Digital Systems and Digital TDM

• The error due to digital coding and transmission
  in a properly functioning system can be
  controlled (and made very small) by the designer
• The quantizing or coding error arising from
  encoding round off should be the only practical
  error in a properly functioning system
• In a properly designed system, the difference in
  signal value (voltage, phase, etc..) for two
  distinct digital symbols is chosen to be much
  larger than any expected but undesired noise
  and interference
• Practical bit error rate (BER) in a good
  telephone system is 10-14
• At 50 Mb/s, one bit error occurs per 555.5 hours
  (23.15 days) on average
• Certain processing is more feasible when the
  signal is represented in digital form
• Digital Signal Processing (DSP), including
  logical processes
• Encryption (where required) is simpler
• Transmission of digital data and digitally coded
  speech should, in principle, permit less costly
  shared facilities (I.e., no modems needed)
• This is one of the motivations for ISDN, although
  the promised cost savings over modem use is not
  fully realized with present-day ISDN

7
Digital Telephone Systems

• Speech quality is equally good regardless of
  geographical distance
• Delay, and thus possibly echo, is the only
  negative consequence of distance, and echo can be
  very effectively reduced to a negligible level
  via echo canceling equipment
• Equipment is superior to analog transmission for
  several reasons
• Lower initial capital and recurring (maintenance)
  costs
• Very compact, high capacity per wire or fiber
• Cross-Fertilization benefits from other digital
  technologies
• Digital switching
• Computers
• Data communications
• all use the same technology, sometimes the exact
  same parts (e.g. memory, logic gates, etc.),
  leading to economy of scale. Design and
  development cost is amortized over a large
  quantity production.

8
Mild Disadvantages of Digital Systems

• More complexity, more components than some cases
  of corresponding analog systems
• Not economically feasible historically with
  vacuum tube hardware
• Integrated circuits make this a much less
  significant disadvantage
• A digitally coded representation of a waveform
  may require more bandwidth for transmission than
  the original analog waveform
• Use of a sophisticated encoding process can
  reduce this problem...
• For example, several low bit rate speech coders
  (8 kb/s or less) use less radio bandwidth for
  cellular and PCS radio than the corresponding
  analog FM radio signal, and produce similar
  perceived speech quality

9
Digital Coding of A Waveform

• Two Major Issues
• Required number of time samples/second
• Required number and distribution of amplitude
  (voltage) samples
• We will consider these issues in that order
• How many samples per second are required to avoid
  missing a short time duration wiggle in the
  waveform?
• How closely spaced must the amplitude quantizing
  levels be to achieve a particular accuracy goal?
• One goal hold the ratio of signal to quantizing
  noise below a specific level.

10
Telephone Voice Bandwidth Previously Standardized

• Result of FDM design studies in 1920s
• 3.5 kHz upper frequency, approximately 300 Hz
  lower frequency. (using 3 dB half power points
  to define bandwidth)
• Lowest frequency is nominally 300 Hz.
• Not high fidelity, which requires 15 kHz to 20
  kHz audio bandwidth, and low frequency response
  down to 20 or 30 Hz.
• Inadequate to recognize some isolated phoneme
  sounds without benefit of known language context
• Examples f, s, sh, th are sometimes confused
• Spelling (names of alphabetic characters) b, d,
  t, even v etc. are sometimes confused
• Requires phonetic alphabet for spoken spelling,
  like ICAO Alpha, Bravo, Charlie, Delta, Echo,
  Foxtrot,
• ICAOInternational Civil Aviation Organization

11
Empirical Telephone Bandwidth

• The nominal 3.5 kHz bandwidth for telephone voice
  connections was established by simple empirical
  testing in 1920s
• Human subjects listened to recordings of
  connected speech statements in their own language
• Various samples were low-pass filtered with upper
  cutoff frequency adjusted
• Percentage of incorrectly perceived samples was
  examined vs. cutoff frequency
• Bandwidth which permitted 99.9 accurate
  perception was used
• Incidentally, narrower 2 kHz bandwidths giving
  only 75 accuracy were used temporarily during
  WW2 to increase link capacities.
• Different low frequency cutoffs (300, 500, 800
  Hz) affect naturalness (presence) of speech,
  but do not affect accuracy of understanding very
  much.
• Existing telephone hardware causes 300 Hz lower
  cutoff, primarily from coupling transformers in
  subscriber loop and microphone/earphone
  limitations.

12
Why use the Narrowest Bandwidth?

• Narrower signal bandwidth permits packing more
  individual channels into a fixed total bandwidth
• This is particularly important in analog FDM
• In digitally coded systems, less bit rate is
  needed to properly code a narrow band signal
  (more later on this point)
• Engineers are usually required to build the most
  economical system which meets quality
  requirements (Barely Adequate system)
• Systems with higher quality requirements use
  greater audio bandwidth
• AM Broadcasting 5 kHz (4.5 kHz in some
  countries)
• FM Broadcasting at least 15 kHz audio bandwidth
• Hi-Fi audio 20 Hz lows and 20 kHz highs (Compact
  Disks)

No standard on low frequency. Most AM
broadcasts roll off at about 100 Hz.
13
The Nyquist Sampling Theorem

• A band-limited waveform can be accurately
  reconstructed if sampled at a rate greater than
  twice its bandwidth.
• Example a 4 kHz bandwidth signal must be sampled
  slightly more than 8000 samples per second
• Exactly 8000 samples/sec would sample each 4000
  Hz sine wave component exactly twice per cycle.
• Theoretical truly band-limited signal has
  absolutely no audio power above some upper
  frequency
• could be produced most practically by a test
  signal generator via adding several sine waves
• Real band-limited voice signal is produced by
  low-pass filtering a real speech source waveform.
  Power above 4 kHz is 30 dB below (1/1000 of) the
  in-band typical power
• Nyquist theorem does not consider amplitude
  quantization errors
• Published by Harry Nyquist of Bell Laboratories
  in 1930s. Nyquist did not consider effects of
  digital quantization, but investigated a
  continuous accurate representation of each sample
  with perfect error-free addition of samples.

14
Recall waveforms can be analyzed into sine waves
Example shows one cycle (T1 second) of a square
wave and lowest three harmonics, sine wave
components.
15
Fourier Analysis

• How big should each sine wave component be? What
  is the appropriate multiplier bk for the k-th
  sine wave
• J.B.Fourier found in 18th century that the
  multiplier can be found from the product integral
• This formula computes the cross correlation
  coefficient between the sine wave and the square
  wave f(t). This is the area on graph paper of
  the product waveform from multiplying an
  appropriate frequency sine wave together with the
  waveform to be analyzed.
• Because the sine wave and this particular example
  square wave have the same so-called odd
  symmetry we do not expect to find a cosine wave
  as well. In general, for non-symmetric waveforms
  f(t), each harmonic term comprises both a cosine
  and sine term.

16
Fourier Coefficients

• From the previous formula and this particular
  square wave we find the first 5 coefficients
• Note that even harmonics (k2, 4,) are all zero.
  This is a special result for a square wave. A
  triangular wave has non-zero even harmonics, for
  example.
• Incidentally, when a non-linear distortion causes
  peak flattening of a waveform, thus making it
  appear more like a square wave, we quantitatively
  measure this by measuring the amount of odd
  harmonics power produced due to the flattening
  of the peaks. This measurement is used
  extensively in high fidelity equipment
  descriptions.

17
Approx. Square Wave Using First Three Odd (1,3,5)
Harmonics
Proper amplitude of each harmonic sine wave was
found from a product integral formula (same as
statistical cross correlation).
18
Example- I

• Consider a waveform like the approximate square
  wave made of only 3 odd-multiple frequency
  harmonics
• The highest frequency sine wave in that example
  was at 5 times the basic periodic frequency
• This synthetic waveform can be generated with
  absolutely no power above a specified upper
  frequency limit
• A filtered real waveform has very little (but
  not zero) power above a specified upper frequency
  limit
• If we can sample that highest sine wave
  frequently enough to capture its amplitude and
  phase precisely, we can reproduce it from the
  sample information
• Sampling exactly 2 times in a cycle is not quite
  enough, but slightly more than 2 samples per
  cycle is OK
• ?t lt t/2, where t (1/fmax) is the period of the
  highest frequency sine wave component.
• No problem to accurately represent lower
  frequency sine wave components using this
  sampling rate

19
Example- II
tT/5 one cycle of 5th harmonic
t/5one cycle
Two sine waves same frequency different
amplitude, phase
?tt/2

• Exactly 2 samples/cycle is ambiguous which sine
  wave is it? This problem
• is called aliasing since more than one sine
  wave (different amplitude and
• phase) fit the same sample points.

20
Example- III

• More than 2 samples/cycle is unambiguous.

?tltt/2




21
Time-Domain Nyquist Rule

• In the time domain, the equivalent rule is that a
  waveform consisting of sine waves can be measured
  at time intervals of ?t and then accurately
  reconstructed if the waveform has no significant
  wiggles (half-period sine wave components)
  which are shorter than the ?t time interval.
• This requires examining the entire time history
  of the waveform, in principle.
• But Fourier analysis of a waveform implies that
  we have examined the entire time history to
  compute the integral products used to evaluate
  the coefficients of the various sine wave terms.
  When we know the amplitude and phase of all the
  frequency components, we can predict the value
  of the sum of all frequency components for any
  time.
• Therefore, the frequency domain statement of
  Nyquists rule implies a complete (time history)
  examination of the waveforms properties.
• Furthermore, telephone engineers were already
  used to measuring the bandwidth of audio
  signals.

22
Band-limited Signals

• Real filtered signals cannot have zero power over
  a non-zero range of frequencies
• OK to have zero power at discrete individual
  frequencies. (for example the case of no even
  harmonic power for square waves)
• ITU-T and other telephone systems standards call
  for the filter to reduce the audio power (above 4
  kHz) to approximately 30 dB below (that is 1/1000
  of) the mid-band power level
• for example, see Bellamy (3rd Ed.) Digital
  Telephony, page 97, Fig. 3.6. Precise limit is 28
  dB below midband audio level
• This implies that noise power from imperfect
  filtration will be of a similar low magnitude
• Observe that the ITU curve is 3dB below the 0dB
  reference level at 3500 Hz frequency
• This 3 dB or half-power point is one of several
  ways to describe the bandwidth of a filter. It is
  easy to measure but not fully descriptive.

23
Amplitude Quantization

• The most obvious initial approach to amplitude
  quantization is to use uniform (linear)
  voltage steps, with enough steps to quantize the
  largest expected amplitude into many small
  intervals
• This is done for musical compact disk (CD)
  digital recording using 16 binary bits,
  corresponding to 65536 distinct fixed voltage
  levels. CD sampling rate is 40 ksamples/sec
• Uniform quantizing is the best encoding for
  signals which will be processed via Digital
  Signal Processing (DSP)
• Arithmetic adding, subtracting, etc. are
  straightforward
• Signals not already represented by uniform
  quantization must be converted before DSP
  processing.
• Yet another special jargon meaning of the word
  linear

24
How many bits?

• 16 bits resolution is much better than is needed
  for telephone purposes.
• Remember, the voice waveform has already been
  band-limited to 3.5kHz bandwidth
• Filter imperfections add about -30 dB noise
  (so-called fold-over noise)
• Carbon microphone is not high-fidelity
• Why bother with extra bits?? They cost more in
  hardware and precision of design and manufacture.
• Empirical listener testing indicates about 12-13
  bits of uniform resolution is adequate
• No perception of degradation in telephone voice
  quality
• Logarithmically compressed (companded) steps at
  low level permit equivalent quality with even
  less bits (in fact, 8)

25
Quantizing Noise (Round off Error)

• Whenever the actual voltage falls between two
  quantized amplitude steps, there is a round off
  error (quantization error)
• The error waveform for a ramp quantized with
  uniform steps is shown in Bellamy (3rd Ed.),
  p.100, Fig.3.9.
• The importance of a mid-tread vs. a mid-riser
  quantizer design is more significant when large
  quantizing steps are used.
• Mid-tread has zero output unless analog input
  exceeds voltage step size, so background noise is
  suppressed, but produces worse quantizing error
  at low voice levels.
• Mid-riser produces worse idle channel noise by
  increasing the miniscule background room noise or
  circuit noise, but has less average quantizing
  noise at low signal levels.
• Quantizing error can be characterized as an
  equivalent additive quantizing noise

mid-tread
Quantizer output code value
Analog voltage
mid-riser
code value
Analog voltage
26
Quantizing Noise

• Unlike random additive noise (Gaussian noise),
  quantizing noise is bounded by the voltage step
  value of the least significant bit and has a
  simpler distribution of amplitude
• Quantizing noise disappears during intervals of
  absolute silence (zero analog input) for
  mid-tread quantizer
• For certain types of testing, artificial
  quantizing noise is produced by instantaneously
  multiplying true random noise by the
  instantaneous magnitude of the audio signal
• The special statistical properties of quantizing
  noise yield a better signal-to-noise ratio than
  ordinary noise
• 56 kb/s V.90 data modems work beyond the
  theoretical Shannon limit on their data rate
  because they are limited by quantizing noise, not
  random (Gaussian) noise

27
Logarithmic Companding

• The human ear exhibits a phenomenon called
  masking
• a noise signal is not perceived as objectionable
  unless it is sufficiently large in relation to a
  desired sound present simultaneously
• Small noises are objectionable in a quiet library
• The same small noise is imperceptible at a rock
  concert!
• This principle is the basis of noise reduction
  systems like the Dolby system for sound
  recording
• The recording audio level is automatically
  increased for soft passages
• The playback level is automatically reduced, to
  match, via an auxiliary control signal, so
  desired signal has the original loudness. In
  Dolby system, this is typically a low frequency
  control signal.
• Therefore, noise added by the recording medium
  (e.g., magnetic tape hiss) is not noticeable
  during soft music intervals
• Dolby systems treat different audio frequency
  bands separately (high frequency is noisiest in
  magnetic tape), and use different types of
  auxiliary signals (Dolby B, C, etc.)

28
Other Companding Stuff

• Analog FM radio of all types (broadcast, analog
  cellular, specialized mobile radio -- SMR, etc.)
  uses amplitude companding to reduce perceived
  audio background noise.
• Low amplitude speech is automatically increased
  in power at the transmit end, reduced again at
  the receiver
• No auxiliary time-varying control signal like
  Dolbys is used, just a uniform preset adjustment
  which shrinks the amplitude scale before
  transmission and stretches the amplitude range
  after reception and detection of the audio
• Syllabic companding in analog telephone systems
  (Bellamy, 3rd Ed. p.116ff) is similar
• These systems have a specified time window to
  compute average audio power (typically 5 to 10
  milliseconds)

29
Logarithmic Instantaneous Companding

• Design objective is uniform ratio of
  instantaneous signal to instantaneous quantizing
  noise, over the range of expected amplitudes
• Achieved by using approximately logarithmically
  spaced quantizing intervals
• Quantizing error amplitude is proportional to the
  difference between adjacent levels, and it is
  then in the same proportion (call this ratio H)
  to the mid-level signal amplitude for each level
• Power is proportional to the square of voltage
  amplitude, so a fixed proportionality ratio (H2)
  holds between instantaneous mid-quantizing-level
  power and quantizing noise
• A small practical problem ideal logarithm is not
  practical for v0, since log(0) is negative
  infinity

30
Practical Logarithmic Companding- Coding

• Two methods to shift the logarithmic function
• µ law Shift to left so it goes through v0 by
  adding a constant to the analog voltage input
• A law Shift up by adding a constant to the code
  value result, then replace a small piece with a
  straight tangent line from the origin to a
  pre-designated low voltage point

µ
A
log(1) is zero
milli
Practical peak voltage is 1.55 V (corresponds to
2mW sine wave_at_600?
31
CODEC Block Diagram
Sample time interval 1/8000 sec or 125 µs
volts
volts
volts
3.5 kHz cutoff
Digital output (serial or parallel) Pulse
Code Modulation (PCM)
ms
Analog Multiplier
Analog- Digital Converter (A or ?- law)
CODER
ms
analog input may contain some power above 3.5 kHz
filtered (smoothed) analog signal
(Pulse Amplitude Modulation- PAM) signal
Low-pass Filters
8 kHz clock pulse train
3.5 kHz cutoff
Digital input (serial or parallel)
Sample and Hold, or Pulse Stretcher (Boxcar) Circu
it
analog input
Digital- Analog Converter (A or ?- law)
DECODER
volts
01011010
volts
v
ms
ms
Example 8 serial bits in 125 µs
ms
32
Mathematical Mu (µ)-law Graphnegative voltage
graph (not shown) is odd-symmetric replica of
this, but -127 code value is modified (explained
later)
Decimal code value
Fraction of full scale
1
127
0
f(v) ln(1 ?(v/1.55))/ln(1?) ?? 255
0.5
63
0
0
0
0.5
1
1.5
2
ln is natural (base e 2.718) logarithm, not
decimal base.
instantaneous positive voltage
33
Mathematical A-law Graphnegative voltage graph
(not shown) is odd-symmetric replica of this
Decimal code value
Fraction of full scale
1
127
f(v) (1 ln(A(v/1.55)))/(1ln(A)) A? 87.6
0.5
63
observe the straight line segment starting here.
Green color on color display
0
0
0
0.5
1
1.5
2
instantaneous positive voltage
34
T-1 (DS-1) TDM Frame
125 ?s or 1/8000 second
F 1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20 21 22 23 24
24 8-bit PCM samples per frame, plus one framing
bit per frame
one time slot
One framing pulse per frame
bit label 1 2 3 4 5 6 7 8
5.18 ?s/slot
Except when common channel signaling is used (in
slot 24 of one link for control of a group of
links), bit 8 is robbed and replaced by a
signaling status bit in all slots during one of 6
frames. Signaling synch is related to a 12 or 24
frame sequence established by the framing bit
pattern.
8000 frames/s 193 bits/frame 1.544 Mbit/s bit
rate 0.647 ?s/bit
35
E-1 (CEPT, MIC) Frame
125 ?s or 1/8000 second
0 1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20 21 22 23 24
25 26 27 28 29 30 31
32 8-bit time slots per frame, normally 30 used
for subscriber PCM, two for synch and signals
one time slot
Slot zero contains synchronizing bit pattern and
some trouble-shooting bit patterns.
Slot 16 contains common channel signaling,
either channel associated condition bits, or CCS7
bit label 1 2 3 4 5 6 7 8
3.9 ?s/slot
8000 frames/s 256 bits/frame 2.048 Mbit/s bit
rate 0.488 ?s/bit
36
Important Distinctions

• Both µ-law and A-law coders use 8 bits for each
  sample
• For international calls, a translation via ROM
  table look-up is done between A and µ (in the µ
  law country)
• When arithmetic operations must be done, the 8
  bit code sample must be converted into a 12 bit
  (or more) sample via a look-up table or other
  means
• 16 bits (with 12 bit accuracy) is also used
• Performing arithmetic directly on companded
  values would not be meaningful
• Not even a precise logarithmic value is used in
  the coding. The result of adding is not the sum
  of two logarithms exactly (although it is
  numerically close for large amplitude values)

37
Other Discrepancies

• Both µ-law and A-law use a sign and magnitude
  representation of the coded value
• Physical zero volts has two codes 0 and -0
• Virtually all computers today use
  twos-complement coding instead to represent
  negative numbers
• Conversion from 8-bit telephonic PCM codes to
  12-bit numeric codes for DSP must correct for
  this as well
• µ-law intentionally modifies the largest negative
  coded value to prevent occurrence of all-zero
  codes after bit inversion occurs for line coding
  (to be explained). A-law does not do this.
• In many transmission systems using µ-law, the 8th
  bit is modified for signaling reasons (to be
  explained) in some frames of data

38
Practical Companding

• The earliest 8-bit uniform Analog/Digital (A/D)
  converters used in D1 version of T-1 systems used
  non-linear logarithmic companding.
• Companding was achieved using an analog
  non-linear amplifier
• Semiconductor diodes have reasonably accurate
  logarithmic relationship between current and
  voltage over part of their operating range. This
  was the technical basis of logarithmic
  companding.
• In contrast, present T-1 and E-1 designs first
  perform 13 bit uniform A/D conversion, then
  produce a companded 8-bit binary number by table
  lookup of an approximate µ-law (or A-law) table.
  (Uniform A/D conversion may use Sigma-Delta
  digitization.)
• This table represents many straight diagonal line
  segments which approximate the smooth µ-law
  formula curve

39
Sign-magnitude vs. 2s Complement

• Sign-magnitude still used in some CDC, Cray, Sun
  super-computers

This value is not used in µ-law voice encoding.