Summary of Sampling, Line Codes and PCM

Prepared for ELE 745Xavier Fernando

Ryerson Communications Lab

Signal Sampling

• Sampling is converting a continuous time signal into a discrete time signal

• Categories:– Impulse (ideal) sampling– Natural Sampling– Sample and Hold operation

Impulse Sampling

Impulse Sampling

• Impulse train spaced at Ts multiplies the signal x(t) in time domain, creating – discrete time, – continuous amplitude signal xs(t)

• Impulse train spaced at fs convolutes the signal X(f) in frequency domain, creating – Repeating spectrum Xs(f) – spaced at fs

The Aliasing Effect

fs > 2fm

fs < 2fm

Aliasing happens

Aliasing

Under sampling will result in aliasing that will create spectral overlap

Ideal Sampling and Aliasing

• Sampled signal is discrete in time domain with spacing Ts

• Spectrum will repeat for every fs Hz

• Aliasing (spectral overlapping) if fs is too small (fs < 2fm)

• Nyquist sampling rate fs = 2fm

• Generally oversampling is done fs > 2fm

Natural Sampling

Natural Sampling

• Sampling pulse train has a finite width τ

• Sampled spectrum will repeat itself with a ‘Sinc’ envelope

• More realistic modeling

• Distortion after recovery depends on τ/Ts

Different Sampling Models

Quantization• Quantization is done to make the signal

amplitude discrete

Analog Signal

Discrete TimeCont. Ampl. Signal

Discrete Time & Discrete

Ampl Signal

Binary Sequence

Linear Quantization

L levels (L-1)q = 2Vp = Vpp

For large LLq ≈ Vpp

PCM Mapping

Linear Quantization Summary• Mean Squared Error (MSE) = q2/12• Mean signal power = E[m2(t)]• Mean SNR = 12 E[m2(t)]/q2

• For binary PCM, L = 2n n bits/sample• Let signal bandwidth = B Hz

– If Nyquist sampling 2B samples/sec– If 20% oversampling 1.2(2B) samples/sec

• Bit rate = 2nB bits/sec• Required channel bandwidth = nB Hz

Non-Uniform Quantization

• In speech signals, very low speech volumes predominates– Only 15% of the time, the voltage exceeds the

RMS value

• These low level signals are under represented with uniform quantization– Same noise power (q2/12) but low signal power

• The answer is non uniform quantization

Uniform Non-Uniform

Non-uniform Quantization

Compress the signal first Then perform linear quantization Result in nonlinear quantization

µ-law and A-lawWidely used compression algorithms

Line Coding• Digital output of the PCM coder is converted

to an appropriate waveform for transmission over channel line coding or transmission coding

• Different line codes have different attributes• Best line code has to be selected for a given

application and channel condition

Line Coded Waveforms - I

NRZ – Non Return to Zero-Level

NRZ – Non Return to Zero-Mark (0no change, 1 change)

NRZ – Non Return to Zero-Space (1no change, 0 change)

Bipolar Return to Zero

AMI – Alternate Mark Inversion (zero zero, 1 alternating pulse)

Bi-Phase level (1 +v-v, 0 -v+v)

Bi Phase Mark

Bi-Phase Space

Delay Modulation

Dicode NRZ

Dicode RZ

1 0 1 1 0 0 0 1 1 0 1

Line Coding Requirements• Favorable power spectral density (PSD)

• Low bandwidth (multilevel codes better)• No/little DC power

• Error detection and/or correction capability• Self clocking (Ex. Manchester)• Transparency in generating the codes (dependency

on the previous bit?)• Differential encoding (polarity reversion)• Noise immunity (BER for a given SNR)

Some Power Spectral Densities

Polar Signalling {p(t) or –p(t)}

• Polar signalling is not bandwidth efficient (best case BW = Rb . Theoretical min is Rb/2)

• Non-zero DC• No error detection (each bit is independent)• Efficient in power requirement• Transparent• Clock can be recovered by rectifying the

received signal

On-Off Signalling

• On-off is a sum of polar signal and periodic clock signal (Fig. 7.2) spectrum has discrete freq. Components

• Polar amplitude is A/2 PSD is scaled by ¼• No error detection• Excessive zeros cause error in timing extraction• Excessive BW• Excessive DC

AMI (bipolar) Signalling

• DC null • Single error detection (violation) capability• Clock extraction possible• Twice as much power as polar signalling• Not transparent• Excessive zeros cause timing extraction error

HDB or B8ZS schemes used to overcome this issue

Bipolar with 8 Zeros Substitution

• B8ZS uses violations of the Alternate Mark Inversion (AMI) rule to replace a pattern of eight zeros in a row.

• 0 0 0 0 0 0 0 0 0 0 0 V 1 0 V 1• Example: (-) 0 0 0 - + 0 + - OR• (+) 0 0 0 + - 0 - +• B8ZS is used in the North American telephone

systems at the T1 rate

High Density Bipolar 3 code

• HDB3 encodes any pattern of more than four bits as B00V (or 100V; 1B (Bit))

• Ex: The pattern of bits1 1 0 0 0 0 1 1 0 0 0 0 0 0+ - 0 0 0 0 + - 0 0 0 0 0 0 (AMI) • Encoded in HDB3 is:+ - B 0 0 V - + B 0 0 V 0 0, which is:+ - + 0 0 + - + - 0 0 - 0 0

M-Ary Coding (Signaling)• In binary coding:

– Data bit ‘1’ has waveform 1– Data bit ‘0’ has waveform 2– Data rate = bit rate = symbol rate

• In M-ary coding, take M bits at a time (M = 2k) and create a waveform (or symbol).– ‘00’ waveform (symbol) 1– ‘01’ waveform (symbol) 2– ‘10’ waveform (symbol) 3– ‘11’ waveform (symbol) 2– Symbol rate = bit rate/k

M-Ary Coding

• Advantages:– Required transmission rate is low (bit rate/M)– Low bandwidth

• Disadvantages:– Low signal to noise ratio (due to multiple

amplitude pulses)

M-ary Signaling

8-level signaling

2-level signaling

M-ary (Multilevel) Signaling• M-ary signals reduce required bandwidth• Instead of transmitting one pulse for each bit

(binary PCM), we transmit one multilevel pulse a group of k-bits (M=2k)

• Bit rate = Rb bits/s min BW = Rb/2

• Symbol rate = R/k sym/s min BW = Rb/2k

• Needed bandwidth goes down by k • Trade-off is relatively high bit error rate (BER)

Inter Symbol Interference (ISI)

• Unwanted interference from adjacent (usually previous) symbols

Nyquist's First Criterion for Zero lSI

• In the first method Nyquist achieves zero lSI by choosing a pulse shape that has a nonzero amplitude at its center (t=0) and zero amplitudes at (t=±nT" (n = I. 2. 3 .... )).

Min. BW Pulse satisfying the first criteria

Zero ISI Pulse

Vestigial Spectrum

Raised Cosine Pulse

r=0 (fx=0)

r=0.5 (fx=Rb/4)

r=1 (fx=Rb/2)

Raised Cosine Filter Transfer Functionin the f domain

Raised Cosine Filter Impulse Response (time domain)

Note pulse rapidly decays for r = 1

Equalization• The residual ISI can be

removed by equalization• Estimate the amount of

ISI at each sampling instance and subtract it

Eye Diagram

• Ideal (perfect) signal

• Real (average) signal

• Bad signal

Eye Diagram

• Run the oscilloscope in the storage mode for overlapping pulses

• X-scale = pulse width• Y-Scale = Amplitude• Close Eye bad ISI• Open Eye good ISI

Time Division Multiplexing (TDM)• TDM is widely used in digital communication

systems to maximum use the channel capacity

Digit Interleaving

TDM – Word Interleaving

TDM

• When each channel has Rb bits/sec bit rate and N such channels are multiplexed, total bit rate = NRb (assuming no added bits)

• Before Multiplexing the bit period = Tb

• After Multiplexing the bit period = Tb/N

• Timing and bit rate would change if you have any added bits

North American PCM Telephony• Twenty four T1 carriers (64kb/s) are multiplexed

to generate one DS1 carrier (1.544 Mb/s)

Each channel has 8 bits – 24 Channels

• Each frame has 24 X 8 = 192 information bits• Frame time = 1/8000 = 125 μs.

T1 System Signalling Format

193 framing bits plus more signalling bits final bit rate = 1.544 Mb/s

North American Digital Hierarchy

Delta Modulation

Why transmit every sample? You know the next amplitude will differ by only ‘delta’

Delta Modulation

Why transmit every sample? You know the next amplitude will differ by only ‘delta’Only transmit the error

LPC Coding • In modern communication system, the voice is artificially generated at the receiver mimicking the original voice using the appropriate coefficients

Transmit only few gain coefficients!

Example -1

Sklar 3.8: (a) What is the theoretical minimum system bandwidth needed for a 10 Mb/s signal using 16-level PAM without ISI?

(b) How large can the filter roll-off factor (r) be if the applicable system bandwidth is 1.375 MHz?

Solution

Example - 2Sklar 3.10: Binary data at 9600 bits/s are transmitted using 8-ary PAM modulation with a system using a raised cosine roll-off filter characteristics. The system has a frequency response out to 2.4 kHz.(a) What is the symbol rate(b) What is the roll o® factor r

Example 3

Sklar 3.11: A voice signal in the range 300 to 3300 Hz is sampled at 8000 samples/s. We may transmit these samples directly as PAM pulses or we may first convert each sample to a PCM format and use binary (PCM) waveform for transmission.

(a)What is the minimum system bandwidth required for the detection of PAM with no ISI and with a filter roll-off factor of 1.

(b) Using the same roll-off, what is the minimum bandwidth required for the detection of binary PCM waveform if the samples are quantized to 8-levels

(c) Repeat part (b) using 128 quantization levels.