comm2

Objective: Analyze Pulse Code Modulation system o

Pulse Code Modulation

PCM was developed by AT&T in 1937 in Paris Laboratories. Alex H. Reeves is credited with its invention. Although PCM were recognized in its early development, it was not until 1960 with the advent of solid state electronics that the PCM became prevalent.

Pulse Code Modulation

PCM was developed by AT&T in 1937 in Paris Laboratories. Alex H. Reeves is credited with its invention. Although PCM were recognized in its early development, it was not until 1960 with the advent of solid state electronics that the PCM became prevalent.

Pulse Code Modulation o

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pulses are of fixed length and fixed amplitude form of digitally coding analog signals a binary system where a pulse or lack of  pulse within a prescribed time slot represents either a logic 1 or a logic 0 condition the analog signal is sampled and then converted to a serial n-bit binary code for transmission. Each code has the same number of bits and requires the same length of time for transmission

Input signal

BPF

Sample and Hold

Regenerative Repeater

Serial to Parallel Converter

DAC

ADC

Parallel to Serial Converter

Regenerative Repeater

Hold Ckt

LPF

Simplified block diagram of a single-channel, simplex PCM transmission system

– limits the frequency of the analog input signal to standard voice band frequency – periodically samples the analog input signal and converts those samples to a multilevel PAM codes – converts the PAM samples to parallel PCM codes – converts the parallel PCM codes to serial digital pulses – regenerate the digital pulse – converts the serial pulses received from the transmission line the parallel PCM codes – converts the parallel PCM codes to multi-level PAM signals – converts the PAM signals back to its original form

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Sampling Quantizing Coding

the function of a sampling circuit in a PCM transmitter is to periodically sample the continually changing analog input voltage and convert those samples to a series of constant amplitude pulses that can more easily be converted to binary PCM code o

for the ADC to accurately convert a voltage to a binary code, the voltage must be relatively constant so that the ADC can complete the conversion before the voltage level changes o

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Flat-top sampling – the sample voltage is held at a constant amplitude during the A/D conversion time; this is done by sampling the analog signal for a short period of time Natural sampling – the sample time is made longer and the analog-to-digital conversion takes place with changing analog signal. This introduces more aperture distortion and requires a faster A/D converter

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PAM pulses generated from natural sampling are not frequently used. Instead, flat-topped pulses are customarily used because flat-top sampling facilitates the design of the electronic circuitry to perform sampling.

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Its purpose is to sample periodically the continually changing analog input signal and convert the samples to a series of constant amplitude PAM levels Sampling p ulse

Analog input

Z1

Z2 Q1 C1

Sample-and-Hold ci rcuit

PAM ou tput

– the time that Q1 (FET) is “ON” and the capacitor charges (or discharges) – the storage time of the capacitor during which the ADC converts the sampling voltage to a PCM code – the distortion that results if the input to the ADC is changing while it is performing the conversion – the gradual discharge of the capacitor during the conversion time

,

τ = RC (to satisfy accuracy requirements)

– output impedance of Z1 plus „ON‟ resistance of Q 1 – capacitance value 10

2.3RC

1

4.6RC

0.1

6.9RC

0.01

9.2RC

to satisfy the output current limitations

i

cdv 

dt 

c

id t  

dv

– max capacitance – max current – maximum change in voltage across C 1 – charge time on aperture time

Example For the sample and hold circuit, determine the largest-value capacitor that can be used if the output impedance of Z 1 = 10Ω, “ON” state resistance of the FET is 10Ω and an acquisition time of 10microseconds. The maximum p-p voltage is 10V and the maximum current from Z1 is 10mA and an accuracy of 1%.

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For a sample to be reproduced accurately at the receiver, each cycle of the analog signal (fa) must be sampled at least twice.

 f s  2 f a o

If fs is less than two times fa, distortion will result. This distortion is called aliasing or foldover distortion.

Nyquist sampling rate for low-pass and bandpass signals   According to the Nyquist theorem, the sampling rate must be at least 2 times the highest frequency contained in the signal.

Recovery of a sampled sine wave for different sampling rates  Sampling at the Nyquist rate can create a good approximation of the original sine wave. Oversampling can also create the same approximation, but is redundant and unnecessary. Sampling below the Nyquist rate does not produce a signal that looks like the original sine wave.

Sampling of a clock with only one hand  The second hand of a clock has a period of 60 s.  According to the Nyquist theorem, we need to sample hand every 30 s

Examples  An example of under-sampling is the seemingly backward rotation of the wheels of a forward-moving car in a movie. A movie is filmed at 24 frames per second. If a wheel is rotating more than 12 times per second, the under-sampling creates the impression of a backward rotation.

Telephone companies digitize voice by assuming a maximum frequency of 4000 Hz. The sampling rate therefore is 8000 samples per second.

Example  A complex low-pass signal has a bandwidth of 200 kHz. What is the minimum sampling rate for this signal?

Solution The bandwidth of a low-pass signal is between 0 and f, where f is the maximum frequency in the signal. Therefore, we can sample this signal at 2 times the highest frequency (200 kHz). The sampling rate is therefore 400,000 samples per second.

f s - f a

2f s - f a 3f s - f a

3f s + f a

Audio f a

0

2f s

No aliasing 2f s - f a

f s - f a

3f s - f a

Shaded areas indicate spectral foldover 3f s + f a

Audio 0

f a

f s

2f s

Aliasing distortion

3f s

o

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process of converting an infinite number of facilities to a finite number of conditions process of rounding off the amplitudes of flat-top samples to a manageable number of levels

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because the codes on the bottom half of the table are a mirror image of the codes on the top half 

1

11

+3

+2.5 V to + 3.5 V

1

10

+2

+1.5 V to + 2.5 V

1

01

+1

+0.5 V to + 1.5 V

1

00

+0

0 V to + 0.5 V

0

00

-0

0 V to - 0.5 V

0

01

-1

-0.5 V to - 1.5 V

0

10

-2

-1.5 V to - 2.5 V

0

11

-3

-2.5 V to - 3.5 V

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o

o

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magnitude difference between adjacent steps

occurs if the magnitude of the sample exceeds the highest quantization interval

assigning PCM codes to absolute magnitudes

also called the step size, is the minimum voltage other than zero volt that can be decoded by the DAC in the receiver equal to the voltage of the least significant bit of the PCM code indicates how many divisions the ADC conversion process uses The smaller the magnitude of the minimum step size the better the resolution and the more accurately the quantization interval will resemble the actual analog sample.

Quantization Error(Qe) or Quantization Noise(Qn) o

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any round off errors in the transmitted signal that are reproduced when the code is converted back to analog in the receiver the maximum amplitude is one-half the resolution

Qe max

1 

2

Vlsb

Qe – maximum Quantization error Vlsb – magnitude of the least significant bit or the step size

Coding

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Dynamic Range (DR) is the ratio of the largest possible magnitude to the smallest possible magnitude that can be decoded by the DAC (Digital-to-Analog Converter) V max V  max V  max  DR   DR   DR 20 log resolution V  min V  min 

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The number of bits used for a PCM code is

2

n

 DR  1

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numerical indication of how efficiently a PCM code is utilized ratio of the minimum bits required to achieve a certain dynamic range to the actual number of bits used

minimum no. of bits Coding efficiency  x100% actual number  of bits

Including the sign bit

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the ratio of signal voltage to quantization noise voltage or the ratio of signal power-to-noise power ratio sometimes called signal-to-distortion ratio or signal to noise ratio

SQR ( dB )  10 .8  20 log ◦

◦

v = rms signal voltage q = quantization interval

v q

Example For a PCM system with the following parameters, determine (a) minimum sample rate (b) minimum number of bits used in the PCM code (c) actual dynamic range (d) resolution (e) quantization error (f) coding efficiency Maximum analog input frequency = 4kHz Maximum decoded voltage at the receiver = ±2.55V Minimum DR = 46 dB Determine the SQR for the following input signal and quantization noise magnitude. 1 Vrms

0.01

2 Vrms

0.02

3 Vrms

0.01

4 Vrms

0.2

Linear codes – the magnitude change between any two successive steps is uniform The accuracy for the higher-amplitude analog signals is the same as for the lower-amplitude signals The SQR for the lower-amplitude signals is less than for the higher-amplitude signals ◦

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Nonlinear codes – the step size increases with the amplitude of the input signal There are more codes for the lower-amplitudes than for higher-amplitudes (for voice transmission) ◦

During times when there is no analog input signal, the only input to the PAM sampler is random, thermal noise. This noise is called idle channel noise and is converted to a PAM sample just as if it were a signal.

– the first quantization interval is made larger in amplitude than the rest of the steps It is a method to reduce idle channel noise – the lowest-magnitude positive and negative codes have the same voltage range as all the other codes

◦

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◦

◦

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This type of coding compares the PAM signal to a ramp waveform while a binary counter is being advanced at a uniform rate Generally limited to low-speed applications This type of coding determines each digit of the PCM code sequentially, it uses a reference weight or code to determine the PCM code Word-at-a-time coders are flash encoders and more complex; they are more suitable for high-speed applications

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Is the process of compressing, then expanding The higher amplitude analog signals are compressed prior to transmission, then expanded at the receiver means of improving the dynamic range of a system Analog

◦





◦

-Law

companding (US and Japan) A-law companding (Europe)

Digital

involves compressing converting it to PCM

first

the

signal

before

 V in     V max  x ln 1   V  max     V out   ln 1    o

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Vmax = max. uncompressed analog input amplitude Vin = amplitude of the input signal at a particular instant of time  = parameter used to defined the amount of  compression Vout = compressed output amplitude

slightly flatter SQR than μ-law but μ-law is better in terms of small signal quality (idle channel noise)

V out   V max

V out   V max

 AV in V max

1  ln  A

0

V in V max

1  ln( AV in V max )

1

1  ln  A

 A





1  A

V in V max

1

Example For a compressor with µ= 255, determine The voltage gain for the following relative values of  (a) Vin: Vmax, 0.75Vmax, 0.5Vmax, 0.25Vmax (b) The compressed output voltage for a maximum input voltage of 4V Input and output dynamic ranges and compression (c) Calculate the output voltage of an A-law compressor given A = 125, Vmax = 8 V and Vin = 25 mV.

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Involves compression at the transmit end after the input sample has been converted to a linear PCM code and expansion at the receive end prior to PCM decoding Sign bit 1=(+) 0=(-)

3-bit Segment identifier  000-111

4-bit quantization interval 0000-1111  ABCD

8-bit 255 compressed code format

Segment

12-bit linear code

8-bit compressed code

8-bit compressed code

12-bit recovered code

Segment

0

S0000000ABCD

S000ABCD

S000ABCD

S0000000ABCD

0

1

S0000001ABCD

S001ABCD

S001ABCD

S0000001ABCD

1

2

S000001ABCDx

S010ABCD

S010ABCD

S000001ABCD1

2

3

S00001ABCDxx

S011ABCD

S011ABCD

S00001ABCD10

3

4

S0001ABCDxxx

S100ABCD

S100ABCD

S0001ABCD100

4

5

S001ABCDxxxx

S101ABCD

S101ABCD

S001ABCD1000

5

6

S01ABCDxxxxx

S110ABCD

S110ABCD

S01ABCD10000

6

7

S1ABCDxxxxxx

S111ABCD

S111ABCD

S1ABCD100000

7

255

encoding/decoding table

% error  

Tx(voltage)   Rx(voltage)  Rx(voltage)

x100

Example Determine the 12-bit linear code, the eight bit compressed code, the decoded 12-bit code, the quantization error, (c) the compression error and (d) percent error for a resolution of 0.01V and analog sample voltages of (a) +0.053V (b) -0.318 V, and (c) +10.234V

% error  

Tx(voltage)   Rx(voltage)  Rx(voltage)

x100

Example Determine the 12-bit linear code, the eight bit compressed code, the decoded 12-bit code, the quantization error, (c) the compression error and (d) percent error for a resolution of 0.01V and analog sample voltages of (a) +0.053V (b) -0.318 V, and (c) +10.234V

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Used when digitizing speech signals only Used primarily in limited bandwidth applications Generally used for recorded information such as “wrong number” messages, encrypted voice for transmission over analog telephone circuits, computer output signals and educational games

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Channel vocoders The first channel vocoder was developed by Homer Dudley in 1928. Dudley‟s vocoder compressed conventional speech waveforms into an analog signal with a total bandwidth of approximately 300 Hz. bandpass filters to separate the speech o Used waveform into narrower subbands. Each sub-band is full-wave rectified, filtered, then digitally encoded. o Operate at 2400 bps o

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Formant vocoders Simply determines the location of the formants and encodes and transmit only the information with the most significant short-term components. o Formants  –  three or more peak frequencies at which the spectral power of most speech energy concentrate o Operate at less than 100 bps o

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Linear predictive vocoders Extracts the most significant portions of speech information directly from the time waveform rather than from the frequency spectrum as with the channel and formant vocoders o Typically transmit and encode speech at between 1.2 and 2.4 kbps o

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simply the data rate at which serial PCM bits are clocked out of the PCM encoder into the Transmission line dependent on the sample rate and the number of bits in the compressed PCM coded

line speed 

samples bits x second sample

Where: line speed – transmission rate in bps o samples/second – sample rate (fs) o bits/sample – number of bits in the compressed PCM code o Example For a single-channel PCM system with a sample rate f s= 6000 samples per second and a seven-bit compressed PCM code, determine the line speed.

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Uses a single-bit PCM code to achieve digital transmission of analog signals If the current sample is smaller than the previous sample, a logic 0 is transmitted If the current sample is larger than the previous sample, a logic 1 is transmitted

Analog input

DAC output





Original Signal

Reconstructed Signal 



Slope overload noise occurs when the step size ∆ is too small for the accumulator output to follow quick changes in the input waveform. Granular noise occurs for any step size, but is smaller for a small step size. If ∆ is decreased, the granular noise will   decrease, however the slope overload noise  will increase.

Thus there should be an optimum value   for the step size ∆.