Digital Transmission Notes

10 Digital Transmission

Transmittal of digital signals between two or more points.
Signals can be binary or discrete-level digital pulses.
Original source can be digital or converted analog.
Requires a physical facility (wires, cable, fiber).
Digital pulses cannot propagate through wireless systems.
First system developed by AT&T for digitally encoded analog signals (voice).
Now used for voice, video, and direct computer data.

10-1-1 Advantages

Noise Immunity: Less susceptible to interference.
- Pulses evaluated during a precise time interval relative to a reference level.
Multiplexing: Better suited for processing and combining signals.
Digital Signal Processing (DSP): Processing analog signals using digital methods (filtering, equalization, phase shifting).
Simpler to store digital signals.
Transmission rate can be easily changed.
Signal Regeneration: More resistant to additive noise using signal regeneration instead of amplification.
- Analog signal-to-noise ratio deteriorates with each amplification.
- Digital regenerators reproduce a new signal with the original signal-to-noise ratio, allowing longer distances.
Simpler to measure and evaluate.
Easier error detection and correction.

10-1-2 Disadvantages

Requires more bandwidth than analog.
Requires analog-to-digital conversion and vice versa.
Requires precise time synchronization.
Incompatible with older analog systems.

10-2 Pulse Modulation

Sampling analog signals and converting samples into discrete pulses.
Four main methods:
- Pulse Width Modulation (PWM) / Pulse Duration Modulation (PDM) / Pulse Length Modulation (PLM).
  - Width of pulse varies with the analog signal amplitude.
  - Maximum amplitude = widest pulse, minimum amplitude = narrowest pulse.
  - All pulses have the same amplitude.
- Pulse Position Modulation (PPM).
  - Position of a constant-width pulse varies within a time slot based on the analog signal sample.
  - Higher amplitude places pulse to the right, lower amplitude to the left.
- Pulse Amplitude Modulation (PAM).
  - Amplitude of a constant-width, constant-position pulse varies with the analog signal sample.
  - Waveforms resemble the original analog signal more than PWM or PPM.
- Pulse Code Modulation (PCM).
  - Analog signal sampled and converted to a serial n-bit binary code.
  - Each code has the same number of bits and transmission time.
PAM is an intermediate form with PSK, QAM, and PCM.
PWM & PPM are for special-purpose (mainly military).
PCM is the most prevalent.

10-3 PCM

Invented by Alex H. Reeves (AT&T, 1937).
Became prevalent in the mid-1960s with solid-state electronics.
Preferred method in public switched telephone networks.
Combines digitized voice and digital data into a single, high-speed digital signal over metallic or optical fiber cables.

PCM Details

PCM is a method of digitally coding analog signals.
Pulses are of fixed length and amplitude.
Binary system: pulse or lack of pulse represents logic 1 or 0.
PWM, PPM, and PAM are digital but seldom binary.
Figure 10-2: Simplified block diagram of a simplex PCM system.
- Bandpass filter limits analog input frequency (300 Hz to 3000 Hz, voice-band).
- Sample-and-hold circuit converts samples to multilevel PAM signal.
- Analog-to-digital converter (ADC) converts PAM samples to parallel PCM codes.
- Parallel-to-serial converter outputs serial binary data onto the transmission line.
- Transmission line repeaters regenerate digital pulses.
- Serial-to-parallel converter converts serial pulses to parallel PCM codes.
- Digital-to-analog converter (DAC) converts parallel PCM codes to multilevel PAM signals.
- Hold circuit (low-pass filter) converts PAM signals back to analog.
Codec (coder/decoder): Integrated circuit performing PCM encoding and decoding.

10-4 PCM Sampling

Sampling circuit periodically samples the analog input voltage and converts samples to constant-amplitude pulses.
ADC requires a relatively constant voltage for accurate conversion.

Techniques

Natural Sampling (Figure 10-3).
- Tops of sample pulses retain their natural shape.
- Difficult for ADC to convert to PCM code.
- Frequency spectrum differs from ideal sample; amplitude decreases for higher harmonics.
- Requires frequency equalizers before recovery by a low-pass filter.
Flat-Top Sampling (Sample-and-Hold circuit).
- Samples input voltage with a narrow pulse and holds it constant.
- Introduces aperture error: amplitude changes during the sample pulse time.
- Magnitude of error depends on analog signal voltage change and sample pulse width.
- Less aperture distortion than natural sampling and allows slower ADCs.

Sample-and-Hold Circuit (Figure 10-5a)

FET (Q1) acts as analog switch.
When on, Q1 provides low-impedance path to deposit analog sample voltage across capacitor C1.
Aperture or acquisition time: Time Q1 is on.
C1 is the hold circuit; when Q1 is off, C1 stores sampled voltage.
A/D conversion time: Time the ADC converts sample voltage to PCM code.
Short acquisition time minimizes change in analog signal while charging C1.
Aperture distortion occurs if the input to the ADC changes during conversion.
Flat-top sampling requires a slower ADC.
Output impedance of voltage follower Z1 and on resistance of Q1 should be small.
Short RC charging time constant allows rapid charging/discharging of the capacitor.
Droop: Gradual capacitor discharge during conversion time due to leakage resistance and input impedance of voltage follower Z2.
High input impedance of Z2 and leakage resistance of C1 are important.
Voltage followers Z1 and Z2 isolate the sample-and-hold circuit from input/output circuitry.
Example 10-1: Calculation of maximum capacitor value for a sample-and-hold circuit based on output current, aperture time and desired accuracy.

10-4-1 Sampling Rate

Nyquist Sampling Theorem: Minimum sampling rate (fs) for accurate reproduction.
- Each cycle of analog input signal (fa) must be sampled at least twice.
- $fs ">= 2fa$ where:
 - $f_s$ = minimum Nyquist sample rate (hertz).
 - $f_a$ = maximum analog input frequency (hertz).
- If $fs < 2fa$ , alias or foldover distortion occurs.

Nonlinear Mixing

Sample-and-hold circuit is a nonlinear device (mixer) with two inputs: sampling pulse and analog input signal.
Frequency-domain representation of the output spectrum includes original inputs, sum and difference frequencies, harmonics, and cross products.
As long as $fs ">= 2fa$ , no aliasing.
Side frequencies from one harmonic spill into sidebands of another harmonic when analog input frequency > $f_s/2$ .
Alias: Frequency that folds over.
Input bandpass filter (antialiasing/antifoldover filter) prevents frequencies greater than one-half the sampling rate from entering the sample-and-hold circuit.
Example 10-2: Determining minimum sample rate and alias frequency for a PCM system with a maximum audio input frequency of 4 kHz.

PCM Binary Codes

Analog input signal is sampled, then converted to a serial binary code.
Binary codes are rc-bit codes.
Codes are sign-magnitude codes (MSB is the sign bit).
Table 10-1: Example of 3-bit PCM code.

10-4-2 Quantization and the Folded Binary Code

Quantization: Converting an infinite number of possibilities to a finite number of conditions.
Analog signals have infinite amplitude possibilities.
Converting analog signal to PCM requires quantization (rounding off).
Total voltage range subdivided into subranges (Table 10-2).
Folded Binary Code: Bottom half of the table is a mirror image of the top half (except for the sign bit).
- Each voltage level has one code, except zero volts (+0 and -0).
Quantization Interval/Quantum: Magnitude difference between adjacent steps.
Maximum signal magnitude is limited by the highest quantization interval.
Overload Distortion/Peak Limiting: Occurs if the sample magnitude exceeds the highest quantization interval.
Quantizing: Assigning PCM codes to absolute magnitudes.
Resolution: Magnitude of a quantum (voltage of the least significant bit - LSB).
Smaller quantum magnitude = better resolution.
Each code has a quantization range equal to $±$ one-half the magnitude of a quantum (except 0-V codes).
Figure 10-8: Analog input signal, sampling pulse, quantized signal (PAM), and PCM code for each sample.
Quantization Error (Qe) / Quantization Noise (Qn): Round-off errors in the transmitted signal, equivalent to additive white noise.
Maximum magnitude for quantization error: one-half a quantum.
Table 10-2 is for a folded PCM code because the codes on the bottom half of the table are a mirror image of the codes on the top half, except for the sign bit.
To determine PCM code, divide the sample voltage by the resolution, convert the quotient to an n-bit binary code, and add the sign bit.
Figure 10-9: PAM signal resembles the analog input signal more closely with faster sampling.
Figure 10-10: Input-versus-output transfer function for a linear analog-to-digital converter.
Example 10-3: Determining quantized voltage, quantization error, and PCM code for a sample voltage.

10-4-3 Dynamic Range

Number of PCM bits per sample is determined by maximum allowable input amplitude, resolution, and dynamic range.
Dynamic Range (DR): Ratio of the largest to smallest magnitude that can be decoded (excluding 0 V).
$DR = \frac{V{max}}{V{min}}$ , where:
- $V_{min}$ = quantum value (resolution).
- $V_{max}$ = maximum voltage magnitude.
$DR = \frac{V_{max}}{resolution}$
Dynamic range is generally expressed in dB: $DR = 20 \log{10} \frac{V{max}}{V_{min}}$
Relationship between dynamic range and number of bits in a PCM code:
- $2^n - 1 \geq DR$
- $2^n — 1 = DR$
n = number of bits in a PCM code (excluding the sign bit).
One positive and one negative PCM code is used for 0 V, which is not considered for dynamic range.
Derivation to solve for the number of bits required, n: n = log(DR + 1) / log 2
Expressing DR in decibels: $DR(dB) = 20 \log_{10}(2^n - 1)$
- Approximation when n > 4: $DR(dB) \approx 6n$
Approximately 6 dB dynamic range for each magnitude bit in a linear PCM code.
Table 10-3 Dynamic Range versus Number of PCM Magnitude Bits.
Example 10-4: Determining minimum sample rate, minimum number of bits, resolution, and quantization error for a PCM system.

10-4-4 Coding Efficiency

Numerical indication of how efficiently a PCM code is utilized.
$\text{Coding Efficiency} = \frac{\text{Minimum Number of Bits}}{\text{Actual Number of Bits}} \times 100$
Example 10-4: Coding efficiency calculation.

10-5 Signal-to-Quantization Noise Ratio

Linear codes (uniform magnitude change between steps).
Maximum quantization noise: half the resolution (quantum value).
Worst-case signal voltage-to-quantization noise voltage ratio (SQR) occurs at minimum amplitude.
$SQR = \frac{Resolution}{Q_n} = \frac{Resolution}{Resolution/2}=2$
$SQR(dB) = 20 \log_{10}(2) = 6 \,\text{dB}$
For a maximum amplitude input signal: $SQR(dB) = 20 \log_{10}(6) = 15.6 \,\text{dB}$
Magnitude of quantization error remains constant, but percentage error decreases as sample magnitude increases.
In reality, the signal power-to-average noise power ratio is generally expressed as:
$SQR(dB) = 10 \log{10} \frac{v^2 / R}{q^2 / 12 / R} = 20 \log{10} \frac{v}{q}$

10-6 Linear Versus Nonlinear PCM Codes

Early PCM systems used linear codes (uniform step magnitude).
Linear coding: Accuracy is the same for higher and lower amplitudes.
Nonlinear/Nonuniform Encoding: Step size increases with the amplitude of the input signal.
More codes for lower amplitudes increase accuracy where needed.
Fewer codes for higher amplitudes increase quantization error (decreasing SQR).
Compromise: Sacrifice SQR for higher amplitudes to achieve more accuracy for lower amplitudes and a larger dynamic range.
Figure 10-11: Step outputs from linear and nonlinear ADCs.
Nonlinear encoding increases the dynamic range.

10-7 Idle Channel Noise

During no-signal times, only random thermal noise is input.
This noise is quantized by the ADC.
Midtread Quantization: Reduces idle channel noise by making the first quantization interval larger (Figure 10-12).
- Allows larger noise to be quantized as + or - zero.
- Disadvantage: larger possible magnitude for Qe in the lowest quantization interval.
Lowest magnitude positive and negative codes have the same voltage range as all the other codes.
Midrise quantization.
Folded binary PCM code eliminates residual noise that fluctuates around 0 V.

10-8 Coding Methods

Methods classified by:
- level-at-a-time.
- digit-at-a-time.
- word-at-a-time.

10-8-1 Level-at-a-Time Coding

Compares PAM signal to ramp while a binary counter advances.
If ramp >= PAM signal, counter contains the PCM code.
Requires fast clock if the PCM code has a large number of bits.
Generally limited to low speed applications.
Nonuniform coding achieved by using a nonlinear ramp.

10-8-2 Digit-at-a-Time Coding

Determines each digit of the PCM code sequentially.
Analogous to a balance using known reference weights.
Provides a compromise between speed and complexity.
Feedback coder uses a successive approximation register (SAR).

10-8-3 Word-at-a-Time Coding

Flash encoders.
Complex but suitable for high-speed applications.
Uses multiple threshold circuits.
Logic circuits sense the highest threshold circuit sensed by the PAM input signal and produce the approximate PCM code.

10-9 Companding

Companding: Compressing and then expanding.
Higher-amplitude signals are compressed before transmission and expanded in the receiver.
Improves the dynamic range of a communications system.
Figure 10-13: Illustration of companding process.
With PCM, companding may be accomplished using analog or digital techniques.

10-9-1 Analog Companding

Historically, used specially designed diodes.
Figure 10-14: Analog companding process.
Transmit side: analog signal compressed, sampled, then converted to a linear PCM code.
Receive side: PCM code converted to PAM signal, filtered, then expanded.
Voice-quality telephone signals require relatively constant SQR.

Log-PCM Codes

Require a logarithmic compression ratio (infinite dynamic range and number of PCM codes - impossible).
Two methods of analog companding in use:
- $\mu$ -Law Companding.
- A-Law Companding.

10-9-1-1 $\mu$ -Law Companding

Used in the United States and Japan.
Compression characteristics for $\mu$ -law:
- $V{out} = V{max}\frac{\ln(1 + \mu \frac{V{in}}{V{max}})}{\ln(1 + \mu)}$ , where:
 - $V_{max}$ = maximum uncompressed analog input amplitude (volts).
 - $V_{in}$ = amplitude of the input signal (volts).
 - $\mu$ = parameter used to define the amount of compression (unitless).
 - $V_{out}$ = compressed output amplitude (volts).
Figure 10-15: Compression curves for several values of $\mu$ .
The higher $\mu$ , the more compression. $\mu$ = 0 => linear (no compression).
$\mu$ determines the range of signal power in which the SQR is relatively constant.
Voice transmission requires > 40 dB dynamic range.

Bell Systems PCM systems

Early: 7-bit code & $\mu$ = 100.
Recent: 8-bit code, $\mu$ = 255.
Example 10-5: Determining voltage gain, compressed output voltage, input and output dynamic ranges, and compression for a compressor with $\mu$ = 255.
Companded systems: compression and expansion factor, which should equal one for all input voltage levels.

10-9-1-2 A-Law Companding

Established by ITU-T for use in Europe.
slightly flatter SQR than $\mu$ -law.
Inferior to $\mu$ -law in small-signal quality (idle channel noise).
Compression characteristic for A-law:
- $V{out} = V{max} \frac{A \frac{V{in}}{V{max}}}{1 + lnA}$ , for $0 \leq \frac{V{in}}{V{max}} \leq \frac{1}{A}$
- $V{out} = V{max} \frac{1 + Ln(A \frac{V{in}}{V{max}})}{1 + LnA}$ , for $\frac{1}{A} \leq \frac{V{in}}{V{max}} \leq 1$

10-9-2 Digital Companding

Compression in the transmitter after the input sample has been converted to a linear PCM code and then expansion in the receiver prior to PCM decoding.
Figure 10-16: Block diagram for a digitally companded PCM system.
Most recent PCM systems use 12-bit linear PCM code and an 8-bit compressed PCM code.
Digital compression closely resembles analog $\mu$ -law curves with $\mu$ = 255 by approximating the curve with eight straight-line segments (segments 0 through 7).
$p$ 255 Encoding algorithm for a 12-bit linear-to-8-bit code is:
- The eight-bit compressed code consists of sign bit, a 3-bit segment identifier, and a 4-bit magnitude code (Figure 10-18a).
- In \[p\255\]-encoding table (Figure 10-18b), the bit positions (designated with an X are truncated during compression and subsequently lost.
Sign bit is transmitted as is.
For segments 0 and 1, the encoded 12-bit PCM code is duplicated exactly at the output of the decoder
With 11 magnitude bits, there are 2048 possible codes, but they are not equally distributed among the eight segments.
Compression process for eight segments for the number of code(s).
Quantization process and segment sub-devision.
In the decoder, the most significant of the truncated bits is reinserted as a logic 1.
- The remaining truncated bits are reinserted as 0.
Example 10-6: Determining the 12-bit linear code, eight-bit compressed code, decoded 12-bit code, quantization error, and compression error for a sample analog voltage.
%error is not equal in all samples but magnitude of error remains constant for equal samples.
Figure 10-19: 12-bit segments divided into subsegments.

10-9-3 Digital Compression Error

The magnitude of the compression error is not the same for all samples.
$\text{Percentage Error} = \frac{\text{12 - bit encoded voltage - 12-bit decoded voltage}}{\text{12-bit decoded voltage}} \times 100$
Example 10-7: Calculation of the percentage error.
Economical method for 12 to 8 and 8 to 12 conversion: Uses a lookup table in ROM (read-only memory).
All functions are performed in single integrated-circuit chip codec.
Most the codecs are called combo chips: Include an antialiasing (bandpass) filter, a sample-and-hold circuit, and an analog-to-digital converter in the transmit section and a digital-to-analog converter, a hold circuit, and a bandpass filter in the receive section.

10-10 VOCODERS

Voice encoders/decoders for speech signals.
Reproduce the short-term power spectrum.
Time waveforms only vaguely resemble the original input signal.
Vocoders cannot be used in applications where analog signals other than voice are present.
Vocoders typically produce unnatural sounding speech.
Vocoders are used primarily in limited bandwidth applications.

10-10-1 Channel Vocoders

Developed by Homer Dudley in 1928.
Compressed conventional speech waveforms into an analog signal with a total bandwidth of approximately 300 Hz.
Present-day digital vocoders operate at less than 2 kbps.
Digital channel vocoders: Use bandpass filters to separate the speech waveform into narrower subbands. Each subband is full-wave rectified, filtered, and then digitally encoded.

Quality

Intelligible but slightly synthetic sounding.

10-10-2 Formant Vocoders

Take advantage of the fact that the short-term spectral density of typical speech signals seldom distributes uniformly across the entire voice-band spectrum (300 Hz to 3000 Hz).
Spectral power concentrates at three or four peak frequencies called formants.
This determine

10-10-3 Linear predictive Coders

Extract the most significant portions of speech information directly from the time waveform rather than from the frequency spectrum.
Linear predictive coder produces a time-varying model of the vocal tract excitation and transfer function directly from the speech waveform.
Natural sounding than either channel or formant vocoder.
####10-11 PCM Line Speed
Data rate at which serial PCM bits are clocked out of the PCM encoder onto the transmission line.
$Line Speed = \frac{Samples}{\text{second}}\ \times\ \frac{\text{bits}}{\text{Sample}}$
*Samples are in bits per sample
*Transmission rate is in bits per second
Line speed is dependent on the sample rate and the number of bits in the compressed PCM code.
*Samples are compressed in PCM code.

10-12 DELTA MODULATION PCM

Uses a single-bit PCM code to achieve digital transmission of analog signals.
With conventional PCM, each code is a binary representation of both the sign and the magnitude of a particular sample. Therefore, multiple-bit codes are required to represent the many values that the sample can be.
With Delta modulation, each transmit represents sample is larger or smaller than the sample before.

10-12-1 Delta Modulation Transmitter

Figure 10-20: Delta Modulation Transmitter Block Diagram.
*Algorithm is quite simple
*A block diagram of the Delta modulation transmitter is also quite simple.
*The diagram is implemented at the block level diagram.
*Samples are larger/smaller which will cause the diagram to go up and down.
*The sampling is up or down depending on if the signal voltage is above or below.

10-12-2 Delta Modulation Receiver

The receiver is almost identical to the transmitter except for the comparator.
Slope Overload
Occurs when the analog input signal changes at a faster rate than the DAC can maintain.
Increasing the clock frequency reduces the probability.
Or increase the magnitude of the minimum step size.
*Another problem is: Granular Noise
Granular Noise: Original signal amplitude is relatively constant, the reconstructed signal has variations that were not present in the original.
Reduce granular noise when decreasing the step size smaller and more detailed samples.
*Granular Noise Is more prevalent in analog signals that have gradual slopes and those amplitudes vary only a small amount.
*Slope Overload is more prevalent in analog signals that have steep slopes or those amplitudes very rapidly.

10-13 Adaptive Delta Modulation PCM

Adaptive delta modulation: Delta modulation system where the step size of the DAC is automatically varied depending on the amplitude characteristics of the analog input signal.
Figure 10-25: shows adaptive modulation structure.
To prevent the possibility of slope overload, increase DAC in number of consecutive of 1s or 0s.

10-14 Differential PCM

With Differential PCM; The differences in amplitude rather than the samples them selves are what is transfer. The individual sample is rarely transferred due to redundancy.
Figure 10-26: DPCM Transmitter.
The samples are then PAM Encoded and transmitted from their respective samples and then their corresponding voltage.

*Transmitter/Receiver can be run in either the Analog or Digital, it depends.

10-15 Pulse Transmission

All Digital carrier systems involve the transmission of pulses and those pulses have a finite bandwidth. A higher more selective system requires a large number of filter sections, which is in practical.
For a transmission signal to properly translate it can be confined:
$B= 1/ T$
To pass the the wave in it's respective form.
*

Nyquist Rate

The maximum signally rate archived through a low- pass filter given has;
$R = 2B$
R has signaling rate.
B = specified bandwidth.

10-15-1 Intersymbol Interference

It's most import in transmission in transmission of pulses with limited band width and nonlinear phase response.
Equalizers are inserted to reduce interference for all frequencies to insure good connectivity.

Timing Inaccuries

Timing errors occur and intersymbol if their respective frequencies line up.
Insufficient Bandwidth
Timing error less likely as transmission rate is far below bandwidth.

Amplitude Distortion

When frequency of communication deviates from expected or original. And then produces the errors

Phase Distortion

Delay Equalizer

10-15-2 Eye Patterns

When displaying and regenerative signals they often form into an eye in the output signal.
An m-level system, there will be m - 1 separate eyes.

Sampling

The mathematicals from all these different sample comes out too;
$ISI = 20 Log (H/ h)$

10-16 Signal Power in Binary Digital Signals

Binary digital Signals can originate from various different data signals
Power can average overall message overall duration, and the signal can be modeled as a connection of a sequence of 1s or 0s.