This document discusses analog and digital signals and pulse code modulation (PCM) systems for digital transmission. It covers key topics such as: 1) Analog signals are continuous over time and amplitude, while digital signals are discrete. PCM allows analog signals to be represented and transmitted digitally via sampling, quantizing, and encoding. 2) The sampling theorem states an analog signal can be reconstructed if sampled at twice its bandwidth. PCM systems sample voice bandwidth signals at 8 kHz and quantize samples with 8-bit coding. 3) Digital transmission provides advantages over analog like reduced noise sensitivity and lower maintenance costs. PCM and techniques like ADPCM and regenerative repeating allow transmission of voice and data over

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INFORMATION: ANALOG AND DIGITAL
David Falconer & Halim Yanikomeroglu
Dept. of Systems and Computer Engineering
Carleton University

2. 2
Topics to be Covered
 Analog (continuous time, continuous amplitude) signals
 Analog to digital: PCM (pulse code modulation)
 Digital transmission

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Analog Signals
Analog (continuous-time, continuous-amplitude) signals (like
speech) have a certain bandwidth. Their power spectrum (power
spectral density) describes how their average power is
distributed with respect to frequency.
Power
spectral
density
(watts/Hz)
0 1 2 3 4 5 6 7....
“High-fidelity speech
Telephone speech
(limited by filtering)
Bandwidth

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Digital and Analog Signals
Some signals (like speech and video) are inherently analog; some
(like computer data) are inherently digital.
However both analog and digital signals can be represented and
transmitted digitally.
Advantages of digital:
» Reduced sensitivity to line noise, temp. drift, etc.
» Lower maintenance costs than analog.
» Low cost digital VLSI for switching and transmission.
» Uniformity in carrying voice, data, video, fax, etc.
» Better encryption.

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Pulse Code Modulation (PCM)
Key points
» PCM signal is developed by three steps: sampling, quantizing and
encoding.
» Quantizing noise is reduced by using variable sized steps. It is independent
of line length.
s(t) s(n)
Sample at t=n Quantize Encode
011010001...
Filter

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Sampling an Analog Signal
Sampling theorem: The original analog signal can be reconstructed if it is
sampled at a rate at least twice its bandwidth.
Reconstruction is by filtering samples with a low pass filter.
Sampling Samples Reconstruction

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Standard PCM in Wired Telephony
Voice circuit bandwidth is 3400 Hz.
Sampling rate is 8 KHz (samples are 125 s apart).
Each sample is quantized to one of 256 levels.
Each quantized sample is coded into a 8-bit word.
The 8-bit words are transmitted serially (one bit at a time) over a
digital transmission channel. The bit rate is 8x8,000 = 64 Kb/s.
The bits are regenerated at digital repeaters.
The received words are decoded back to quantized samples, and
filtered to reconstruct the analog signal.

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Quantization
Uniform Nonuniform
Input signal Input signal
Output signal Output signal
The more steps (levels) the less quantization noise. Nonuniform quantization
(e.g. -law) allows a larger dynamic range (important for speech).

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-Law Quantization and Coding
Standardized in North America.
Based on a logarithmic non-uniform quantizer.
Range of amplitudes divided into 8 segments, each segment with
16 uniformly spaced levels. Segment i is double the width of
segment i-1.
8 bit word: 1 bit for sign, 3 bits identify segment, 4 bits identify level
within segment.
Can show for n-bit word, signal to quantization noise ratio is
approximately 6n-10 [dB]; e.g., 38 dB for n=8 bits.
Most of the rest of the world uses a related logarithmic non-
uniformity, called A-law.

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1 2 3 4 24
1 2 3 4 5 6 7 8
S bit
24 PCM code words, each representing 1 sample
8 bits per code word
193 bits in 125 s
(1.544 Mb/s)
DS1 Format (-Law Countries)

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Adaptive Differential PCM (ADPCM)
Allows coding with a lower bit rate (with same fidelity) for speech,
based on predicting the next sample; e.g., 8 or 16 or 32 Kb/s.
More circuits accommodated in the same transmission bandwidth.
Quant.
Predictor
Predictor
+
+ +
Coder: Decoder:

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Regenerative Repeater
Regenerative
repeater
Regenerative
repeater
Amplifier/
equalizer
Regenerator
Timing circuit
Structure of a regenerative
repeater:
By appropriate repeater design and inter-repeater spacing, the effect of
occasional bit errors due to noise can be controlled. Received signal quality is
essentially independent of distance.

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PCM Transmission Formats and Spectra
..... 1 0 1 1 .......

Time Frequency
0 T 2T 3T -4/T -1/ -2/T -1/T 0 1/T 2/T 1/ 4/T
0 T 2T 3T 4T -3/T -2/T -1/T 0 1/T 2/T 3/T
0 T 2T 3T 4T -4/T -2/T -1/T 0 1/T 2/T 4/T
0 T 2T 3T 4T -1/2T 1/2T
Min. bandwidth
Unipolar RZ
Unipolar NRZ
Bipolar NRZ
Bandlimited
Power spectra

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Multilevel Transmission
1 0 1 1 0 0 0 1
0 T 2T 3T 4T
Binary:
L=2
4-level:
L=4
Bit rate =
1
T
log2 L
Bandwidth proportional to 1/T for NRZ signals

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Bandwidth Required for Digital Transmission
 required bandwidth is approximately
(bit rate)/(log2L) for L-level transmission.
 more levels  less bandwidth, but greater sensitivity to noise.
 Examples:
» 64 Kb/s PCM requires about 64 KHz for binary transmission, 32 KHz for 4-
level transmission.
» 14.4 Kb/s modem uses a symbol rate 1/T=2400 Hz, and the equivalent of
L=32.

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Channel Capacity
Shannon channel capacity formula:
» Highest possible transmission bit rate R, for reliable communication in a
given bandwidth W Hz, with given signal to noise ratio, SNR, is
R=Wlog2(1+SNR) bits/s
R/W = 0.332 SNR [dB] bits/s/Hz (for high SNR)
» Assumptions and qualifications:
– Gaussian distributed noise added to the signal by the channel, highly complex
modulation, coding and decoding methods.
– In typical practical situations, the above formula may be roughly modified by
dividing SNR by a factor of about 5 to 10.

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Summary
 All information signals can be represented, switched, stored and
transmitted digitally.
 We have discussed PCM systems and their key elements:
» sampling
» quantizing
» coding
» digital transmission
 We have discussed the related concepts of:
» the telephone set
» bandwidth
» the sampling theorem
» signal to quantization noise ratio
» channel capacity.

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More Information
R. Haughton, “The Telecommunications Mosaic”, Vol. 2, (sections II.1, II.2, II.3, II.4),
Vol. 3, (sections I.1, I.2, I.3, I.4)
E.B. Carne, “Telecommunications Primer”, Prentice-Hall, 1995, Chapters 2,3,4.
R.L. Freeman, “Telecommunications System Engineering”, (2nd ed.), Wiley, 1989.
Chapter 9.
J. Sklar, “Digital Communications”, Chapters 2 and 7