Time-division multiplexing (TDM)
The method of combining several sampled signals in a
definite time sequence is called time-division multiplexing
(TDM).
TDM for PAM signals
Suppose we wish to time-multiplex two signals using PAM.
Digital logic circuitry is usually employed to implement the
timing operations.
f1 (t )
Sampler
+
f 2 (t )
Sampler
+
Pulse
generator
LPF
PAM (TDM)
Clock
Pulse
generator Commutator
J.1
The time-multiplexed PAM output is
T
f1 (t )
f 2 (t )
Tx
t
Sampling rate
The sampling rate depends on the bandwidth of the signals.
For example, if the signals are low-pass and band-limited to
3kHz. The sampling theorem states that each must be
sampled at a rate no less than 6kHz. This requires a 12kHz
minimum clock rate for the two-channel system.
J.2
Transmission bandwidth
– The time-multiplexed PAM signal can be sent out on a
line (baseband communications) or used to modulate a
transmitter (passband communications).
Theoretically, the bandwidth occupied by a pulse is infinite.
J.3
However, we are transmitting the information of the signals
( f1 (t ), f 2 (t ) ), not the information of the pulses.
If the time spacing between adjacent samples is Tx (In this
example, Tx = T / 2 ), the minimum bandwidth is
Bx = 1 /( 2Tx ) .
J.4
For example, if the time-multiplexed PAM signal described
in J.4 is filtered with a low-pass filter with bandwidth
Tx = 1 /( 2 Bx ) , the impulses become sinx/x terms.
1 /(2 Bx )
t
LPF
t
Because we have chosen the spacing between successive
samples to be 1 /( 2 Bx ) , contributions from all adjacent
channels are exactly zero at the correct sampling instant.
Therefore, by sampling the output at the correct instant, one
can exactly reconstruct the original sampled values
J.5
T
f1 (t )
f 2 (t )
t
LPF
t
Tx
The results refer to the case in which impulse sampling and
ideal filtering. In practice, neither of these conditions can be
achieved and wider bandwidth is required.
The required bandwidth depends on the allowable cross-talk
(interference) between channels.
J.6
Receiver
f PAM (t )
Sampler
Sampler
LPF
f1 (t )
LPF
f 2 (t )
Pulse
generator
Pulse
generator
Clock
Commutator
Synchronization of the the clock and the commutator in the
time-multiplex receiver can be achieved by sending some
pre-assigned code which, when identified at the receiver,
serves to synchronize the timing.
J.7
After time multiplexing and filtering, the pulse-modulated
waveform may be transmitted directly on a pair of wire lines
For long distance transmission, the multiplexed signal is
used as the modulating signal to modulate a carrier.
– For example, PAM/AM
PAM
multiplexer
Clock
AM
modulator
cosωct
AM
demodulator
cosωct
PAM
multiplexer
Clock
J.8
Advantages of TDM
– high reliability and efficient operation as the circuitry
required is digital.
– Relatively small interchannel cross-talk arising from
nonlinearities in the amplifiers that handle the signals in
the transmitter and receiver.
Disadvantages of TDM
– timing jitter
J.9
Example
Channel 1 of a two-channel PAM system handles 0-8 kHz
signals; the second channel handles 0-10kHz signals. The
two channels are sampled at equal intervals of time using
very narrow pulses at the lowest frequency that is
theoretically adequate.
f1(t)
f2 (t)
Sampler
+
LPF
PAM (TDM)
+
Sampler
Pulse
generator
Clock
Pulse
generator
Commutator
J.10
a) what is the minimum clock frequency of the PAM signal ?
The minimum sampling rate for channel 1 is 2B = 16kHz.
The minimum sampling rate for channel 2 is 20kHz.
In order to sample channel 2 adequately, we must take
samples at a 20kHz rate. Therefore the commutator clock
rate is 40kHz.
J.11
b) What is the minimum cutoff frequency of the low-pass filter
used before transmission that will preserve the amplitude
information on the output pulses ?
Bx ≥ 1 /( 2Tx ) = 20kHz
c) What would be the minimum bandwidth if these channel
were frequency multiplexed, using normal AM techniques
and SSB techniques ?
AM: 2*(bandwidth of channel 1) + 2*(bandwidth of
channel 2) = 2*8kHz + 2*10kHz = 36kHz
SSB: bandwidth of channel 1 + bandwidth of channel 2
= 8kHz + 10kHz = 18kHz
J.12
d) Assume the signal in channel 1 is sin(5000πt) and that in
channel 2 is sin(10000πt). Sketch these signals; sketch the
waveshapes at the input to the first low-pass filter, at the
filter output, and at the output of the sample-and-hold circuit
and output of the low-pass filter in channel 2.
0.2ms
0.4ms
sin(5000πt )
t
t
sin(10000πt )
J.13
Sampling period = 1/(2*10kHz)=0.05ms
0.2ms
0.4ms
0.05ms
Multiplexed PAM:
Output of filter:
t
0.05ms
t
t
t
J.14
Output of holding circuit
for channel 2:
t
Output of low-pass filter:
t
J.15
Line coding
Return-to-bias (RB) method
– Three levels are used: 0,1, and a bias level.
– Bias level may be chosen either below or between the
other two levels.
– The waveform returns to the bias level during the last half
of each bit interval.
– The RB method has an advantage in being self-clocking.
PCM code
1
1
1
0
0
1
Example:
1 ==> A volts
0 ==> -A volts
RB
J.16
Unipolar Return-to-zero (RZ) method
– Digit ‘1’ is represented by a change to the 1 level for one-half the
bit interval, after which the signal returns to the reference level for
the remaining half-bit interval.
– Digit ‘0’ is indicated by no change, the signal remaining at the
reference level.
– Its disadvantage is that it requires 3dB more power than RB
signaling (or AMI) for the same probability of symbol error.
– An attractive feature of this line code is the presence of delta
function at f=1/Tb in the power spectrum of the transmitted signal,
which can be used for bit-timing recovery at the receiver.
J.17
Tb :Bit duration
PCM code
RZ
1
1
1
0
0
1
Decision boundary
RB
J.18
– power spectrum of Unipolar RZ signaling. The
normalized frequency is 1/Tb
J.19
Alternate Mark Inversion (AMI)
– The first ‘1’ is represented by +1, the second ‘1’ by -1,
the third ‘1’ by +1, etc.
– has zero average value and relatively insignificant lowfrequency components
– used in telephone PCM systems.
– Also referred to as a bipolar return-to-zero (BRZ)
representation.
PCM code
1
1
1
0
0
1
AMI
J.20
– Power spectrum of AMI signaling
J.21
Spilt phase
– eliminates the variation in average value using symmetry.
– In the Manchester split-phase method
• A ‘1’ is represented by a 1 level during the first half-bit interval,
then shifted to 0 level for the latter half-bit interval
• A ‘0’ is indicated by the reverse representation.
– The manchester code suppresses the DC component and has
relatively insignificant low-frequency components.
– In the split-phase (mark) method, a similar symmetric
representation is used except that a phase reversal relative to the
previous phase indicates a ‘1’ and no change is used to indicate a
‘0’.
J.22
PCM code
1
1
1
0
0
1
Split-phase
(Manchester)
Split-phase
(mark)
J.23
– Power spectrum of Manchester code signaling
J.24
Nonreturn-to-zero
– reduce the bandwidth needed to send the PCM code.
– In the NRZ(L) representation, a bit pulse remains in one of its two
levels for the entire bit interval.
– In the NRZ(M) method a level change is used to indicate a ‘1’ and
no level change for a ‘0’.
– In the NRZ(S) method a level change is used to indicate a ‘0’ and
no level change for a ‘1’.
– NRZ representations require added receiver complexity to
determine the clock frequency.
PCM code
1
1
1
0
0
1
NRZ (L)
NRZ (M)
NRZ (S)
Delay Modulation
(Miller code)
J.25
Delay modulation (Miller code)
– a ‘1’ is represented by a signal transition at the midpoint
of a bit interval. A ‘0’ is represented by no transition
unless it is followed by another ‘0’, in which case the
signal transition occurs at the end of the bit interval.
PCM code
1
1
1
0
0
1
NRZ (L)
NRZ (M)
NRZ (S)
Delay Modulation
(Miller code)
J.26
– Power spectrum of NRZ(L)
J.27
Transmission bandwidth
– The fundamental frequency of a binary code stream depends on its most rapidly varying
pattern.
– Example: ‘111’ for RZ and NRZ(M)
1
1
Tb
1
1
1
Tb
f o = 1 / Tb
1
f o = 1 / 2Tb
– For a binary PCM system with n quantization levels, the number of bits per sample is
[log 2 n]
(the brackets indicate the next higher integer to be taken, e.g. if n=7, we use 3 bits)
– If the sample rate be 1/T, then the number of bits per second to be sent is
[log 2 n] / T
– The minimum bandwidth is
B≥
1  [log 2 n] 


2 T 
(NRZ)
B≥
[log 2 n]
T
(RZ)
J.28
– In baseband transmission, the bit stream described in N.1-N.8 are sent on a transmission
line.
– In passband transmission, the bit stream is used to modulate a high frequency carrier.
• Amplitude-shift keying (ASK): the amplitude of a carrier is switched between two
values in response to the PCM code.
• Frequency-shift keying (FSK): the frequency of a carrier is switched between two
values in response to the PCM code.
• Phase-shift keying (PSK): the phase of a carrier is switched between two values in
response to the PCM code.
PCM code
1
1
1
0
0
1
NRZ (L)
ASK
FSK
PSK
change of phase
J.29
– PSK and FSK are preferred to ASK signals for passband
data transmission over nonlinear channel such as
micorwave link and satellite channels.
Coherent and Noncoherent
– Digital modulation techniques are classified into coherent
and noncoherent techniques, depending on whether the
receiver is equipped with a phase-recovery circuit or not.
– The phase-recovery circuit ensures that the local oscillator
in the receiver is synchronized to the incoming carrier
wave (in both frequency and phase).
J.30
Coherent PSK
The functional model of passband data transmission system is
mi
Signal
si
transmission
encoder
Modulator
si (t )
Channel
x(t )
Detector
x
Signal
transmission
m̂
decoder
Carrier signal
• mi is the binary sequence.
– In a coherent binary PSK system, the pair of signals s1 (t )
binary symbols 1 and 0, respectively, is defined by
2 Eb
s1 (t ) =
cos(2πf ct )
Tb
s2 (t ) =
where
and s2 (t )
used to represent
2 Eb
2 Eb
cos(2πf ct + π ) = −
cos(2πf ct )
Tb
Tb
0 ≤ t ≤ Tb , and
Eb is the transmitted signal energy per bit.
J.31
For example,
E=
∫
Tb
0
[s1 (t )] dt = 2 Eb
Tb
2
∫
Tb
0
cos 2 (2πf ct )dt =
2 Eb Tb
⋅ = Eb
Tb 2
To ensure that each transmitted bit contains an integral number of cycles of the carrier
wave, the carrier frequency f c is chosen equal to n / Tb for some fixed integer n.
The transmitted signal can be written as
s1 (t ) = Ebφ (t )
and
s1 (t ) =
2 Eb
2 Eb
n
cos(2πf ct ) =
cos(2π t )
Tb
Tb
Tb
∴ s1 (Tb ) =
s2 (t ) = − Ebφ (t )
2 Eb
cos(2nπ )
Tb
where
φ (t ) =
2b
cos(2πf ct )
Tb
0 ≤ t < Tb
J.32
Generation of coherent binary PSK signals
To generate a binary PSK signal, we have to represent the
input binary sequence in polar form with symbols 1 and 0
represented by constant amplitude levels of + Eb and
− Eb , respectively.
• This signal transmission encoder is performed by a
polar nonreturn-to-zero (NRZ) encoder.
• The carrier frequency f c = n / Tb where n is a fixed
integer.
 + Eb
• si = 
 − Eb
input symbol is 1
input symbol is 0

2 Eb
s
(
t
)
cos(2πf c t )
if si = Eb
=
 1
Tb

si (t ) = 
s 2 (t ) = − 2 Eb cos(2πf c t ) if si = − Eb

Tb
J.33
10101
Signal
si
transmission
encoder
Product
Modulator
φ (t ) =
si (t )
2
cos(2πf ct )
Tb
J.34
Detection of coherent binary PSK signals
To detect the original binary sequence of 1s and 0s, we
apply the noisy PSK signal to a correlator. The correlator
output is compared with a threshold of zero volts.
x(t )
X
φ (t )
∫
Tb
0
x1
Decision
device
1 if x1
0 if x1
0
Correlator
J.35
Example: If the transmitted symbol is 1,
2 Eb
x(t ) =
cos(2πf c t )
Tb
and the correlator output is
x1 =
=
Tb
∫
0
x(t )φ (t )dt
Tb
∫
0
2 Eb
2
cos(2πf c t ) ⋅
cos(2πf c t )dt
Tb
Tb
2
= Eb ⋅
Tb
Tb
∫
0
cos 2 (2πf c t )dt
= Eb
Similarly, If the transmitted symbol is 0, x1 = − Eb .
J.36
Delta Modulation (DM) and Differential Pulse Code Modulation (DPCM)
Reference
– Stremler, Communication Systems, Chapter 9.7
Delta Pulse Code Modulation (DPCM)
– In the transmission of messages having repeated sample values, the repeated
transmission represents a waste of communication capability because there is
little information content in the repeated values.
– In DPCM, only the digitally encoded difference between successive sample
values. Therefore, the number of bit can be reduced.
– Example: a picture that has been quantized to 6 bits can be transmitted with
comparable quality using 4-bit DPCM.
J.37
f (t )
LPF
g (t )
f LP (t ) +
-
f delay (t ) ≈ f LP (t − T )
Decoder
DPCM
f LP (t )
f (t )
t
Clock/
Sampler
Quantizerencoder
∫
Decoder
∫
LPF
f delay (t )
t
DPCM
≈ f (t )
g (t ) = f LP (t ) − f delay (t )
t
Range of f (t ) > Range of g (t )
J.38
Delta Modulation (DM)
– In delta modulation (DM), an incoming signal is oversampled (i.e. at a rate much
higher than the Nyquist rate) to purposely increase the correlation between
adjacent samples of the signal.
– The difference between the input and the approximation is quantized into two ± ∆
levels:
 f q (nT ) + ∆ if f (nT + T ) > f q (nT )
f q (nT + T ) = 
 f q (nT ) − ∆ if f (nT + T ) > f q (nT )
f (nT + T )
+
f q (nT )
Quantizer
Encoder
DM
+
+
Delay T
f q (nT + T )
Accumulator
J.39
f (t )
Slope-overload
Idling noise
t
010 110101111101
t
– Disadvantages
• If the input signal level remains constant, the reconstructed DM waveform
exhibits a hunting behavior known as idling noise.
• Slope-overload
J.40