Summary of Sampling Line Codes and PCM Prepared

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Summary of Sampling, Line Codes and PCM Prepared for ELE 745 Xavier Fernando Ryerson

Summary of Sampling, Line Codes and PCM Prepared for ELE 745 Xavier Fernando Ryerson Communications Lab

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

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 Sampling • Impulse train spaced at Ts multiplies the signal x(t) in time

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 > 2 fm fs < 2 fm Aliasing happens

The Aliasing Effect fs > 2 fm fs < 2 fm Aliasing happens

Aliasing Under sampling will result in aliasing that will create spectral overlap

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

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 < 2 fm) • Nyquist sampling rate fs = 2 fm • Generally oversampling is done fs > 2 fm

Natural Sampling

Natural Sampling

Natural Sampling • Sampling pulse train has a finite width τ • Sampled spectrum

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

Different Sampling Models

Quantization Discrete Time & Discrete Ampl Signal Mapping Discrete Time Cont. Ampl. Signal Quantization

Quantization Discrete Time & Discrete Ampl Signal Mapping Discrete Time Cont. Ampl. Signal Quantization Analog Signal Sampling • Quantization is done to make the signal amplitude discrete Binary Sequence

Linear Quantization L levels (L-1)q = 2 Vp = Vpp For large L Lq

Linear Quantization L levels (L-1)q = 2 Vp = Vpp For large L Lq ≈ Vpp

PCM Mapping

PCM Mapping

Linear Quantization Summary • • • Mean Squared Error (MSE) = q 2/12 Mean

Linear Quantization Summary • • • Mean Squared Error (MSE) = q 2/12 Mean signal power = E[m 2(t)] Mean SNR = 12 E[m 2(t)]/q 2 For binary PCM, L = 2 n n bits/sample Let signal bandwidth = B Hz – If Nyquist sampling 2 B samples/sec – If 20% oversampling 1. 2(2 B) samples/sec • Bit rate = 2 n. B bits/sec • Required channel bandwidth = n. B Hz

Non-Uniform Quantization • In speech signals, very low speech volumes predominates – Only 15%

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 (q 2/12) but low signal power • The answer is non uniform quantization

Uniform Non-Uniform

Uniform Non-Uniform

Non-uniform Quantization Compress the signal first Then perform linear quantization Result in nonlinear quantization

Non-uniform Quantization Compress the signal first Then perform linear quantization Result in nonlinear quantization

µ-law and A-law Widely used compression algorithms

µ-law and A-law Widely used compression algorithms

Line Coding • Digital output of the PCM coder is converted to an appropriate

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 –

Line Coded Waveforms - I NRZ – Non Return to Zero -Level NRZ – Non Return to Zero -Mark (0 no change, 1 change) NRZ – Non Return to Zero -Space (1 no change, 0 change) Bipolar Return to Zero AMI – Alternate Mark Inversion (zero zero, 1 alternating pulse)

1 0 1 1 0 0 0 1 1 0 1 Bi-Phase level (1

1 0 1 1 0 0 0 1 1 0 1 Bi-Phase level (1 +v-v, 0 -v+v) Bi Phase Mark Bi-Phase Space Delay Modulation Dicode NRZ Dicode RZ

Line Coding Requirements • Favorable power spectral density (PSD) • Low bandwidth (multilevel codes

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

Some Power Spectral Densities

Polar Signalling {p(t) or –p(t)} • Polar signalling is not bandwidth efficient (best case

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

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

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 B 8 ZS schemes used to overcome this issue

Bipolar with 8 Zeros Substitution • B 8 ZS uses violations of the Alternate

Bipolar with 8 Zeros Substitution • B 8 ZS uses violations of the Alternate Mark Inversion (AMI) rule to replace a pattern of eight zeros in a row. • 0000 000 V 1 • Example: (-) 0 0 0 - + 0 + - OR • (+) 0 0 0 + - 0 - + • B 8 ZS is used in the North American telephone systems at the T 1 rate

High Density Bipolar 3 code • HDB 3 encodes any pattern of more than

High Density Bipolar 3 code • HDB 3 encodes any pattern of more than four bits as B 00 V (or 100 V; 1 B (Bit)) • Ex: The pattern of bits 11000000 + - 0 0 0 (AMI) • Encoded in HDB 3 is: + - B 0 0 V - + B 0 0 V 0 0, which is: +-+00+-+-00 -00

M-Ary Coding (Signaling) • In binary coding: – Data bit ‘ 1’ has waveform

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 = 2 k) 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

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 Signaling 8 -level signaling 2 -level signaling

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

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

Inter Symbol Interference (ISI) • Unwanted interference from adjacent (usually previous) symbols

Nyquist's First Criterion for Zero l. SI • In the first method Nyquist achieves

Nyquist's First Criterion for Zero l. SI • In the first method Nyquist achieves zero l. SI by choosing a pulse shape that has a nonzero amplitude at its center (t=0) and zero amplitudes at (t=±n. T" (n = I. 2. 3. . )).

Min. BW Pulse satisfying the first criteria

Min. BW Pulse satisfying the first criteria

Zero ISI Pulse

Zero ISI Pulse

Vestigial Spectrum

Vestigial Spectrum

Raised Cosine Pulse r=0 (fx=0) r=0. 5 (fx=Rb/4) r=1 (fx=Rb/2)

Raised Cosine Pulse r=0 (fx=0) r=0. 5 (fx=Rb/4) r=1 (fx=Rb/2)

Raised Cosine Filter Transfer Function in the f domain

Raised Cosine Filter Transfer Function in the f domain

Raised Cosine Filter Impulse Response (time domain) Note pulse rapidly decays for r =

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

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 • Ideal (perfect) signal • Real (average) signal • Bad signal

Eye Diagram • Run the oscilloscope in the storage mode for overlapping pulses •

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

Time Division Multiplexing (TDM) • TDM is widely used in digital communication systems to maximum use the channel capacity Digit Interleaving

TDM – Word Interleaving

TDM – Word Interleaving

TDM • When each channel has Rb bits/sec bit rate and N such channels

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 T 1 carriers (64 kb/s) are multiplexed

North American PCM Telephony • Twenty four T 1 carriers (64 kb/s) are multiplexed to generate one DS 1 carrier (1. 544 Mb/s)

Each channel has 8 bits – 24 Channels • Each frame has 24 X

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

T 1 System Signalling Format 193 framing bits plus more signalling bits final bit

T 1 System Signalling Format 193 framing bits plus more signalling bits final bit rate = 1. 544 Mb/s

North American Digital Hierarchy

North American Digital Hierarchy

Delta Modulation Why transmit every sample? You know the next amplitude will differ by

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

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

LPC Coding Transmit only few gain coefficients! • In modern communicatio n system, the

LPC Coding Transmit only few gain coefficients! • In modern communicatio n system, the voice is artificially generated at the receiver mimicking the original voice using the appropriate coefficients

Example -1 Sklar 3. 8: (a) What is theoretical minimum system bandwidth needed for

Example -1 Sklar 3. 8: (a) What is 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

Solution

Example - 2 Sklar 3. 10: Binary data at 9600 bits/s are transmitted using

Example - 2 Sklar 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 k. Hz. (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

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.