1. Modulation and Multiplexing
GMU - TCOM Spring 2001
Class: Feb
Modulation and Multiplexing
Joe Montana
IT Fall 2003
(C) Leila Z. Ribeiro, 2001

2. Agenda Modulation Concept Analog Communication Digital Communication
Digital Modulation Schemes
Error Detection and Correction

3. Modulation

4. Why Modulate Signals?
If we transmit signal through electromagnetic waves, we need antennas to recover them at a remote point.
At low frequencies (baseband), the wavelengths are very large.
Ex. Voice, at approx. 4 kHz, has a wavelength of 75 Km!!
If we “move” those signals to higher frequencies, we can get more manageable antennas.
After receiving the signal, we need to “move” them back to the original frequency band (baseband) through demodulation.
Therefore, you can see the modulation task as “giving wings” to the information message.

5. Modulation – Basic Principles
Modulation is achieved by varying the amplitude, phase or frequency of a high frequency sinusoid.
The initial high frequency sinusoid that will have a parameter modified is called the “Carrier”.
The original message signal (baseband) is called the “Modulating” signal.
The resulting bandpass signal is the “Modulated” signal, which is a combination of the carrier and the original message.

6. Modulation – Basic Principles
Modulating Signal V(t), at baseband(fB)
Action on carrier’s amplitude, frequency or phase
Modulated Signal carrying the information of V(t), bandpass (fC)
Carrier (fC) fC fC

7. MODULATION AND MULTIPLEXING - 1
THIS IS THE WAY INFORMATION IS ENCAPSULATED FOR TRANSMISSION
MULTIPLEXING
THIS IS THE WAY MORE THAN ONE LINK CAN BE CARRIED OVER A SINGLE COMMUNICATIONS CHANNEL
WE WILL BE LOOKING AT MODULATION INITIALLY, BUT WHERE DO MODULATION AND MULTIPLEXING FIT INTO A SYSTEM?

8. MODULATION AND MULTIPLEXING - 2
Fig. 5.1 in text: (A) At uplink earth station (B) At downlink earth station

9. MODULATION AND MULTIPLEXING - 3
KEY POINTS
You have to multiplex before modulating on the transmit side (that is, you have to get all of the output signals together prior to modulating onto a carrier)
You have to demodulate before demultiplexing on the receive side (that is, before you can separate - i.e. demultiplex - the incoming signals, you have to demodulate the carrier to obtain the transmitted information)

10. Analog Communications

11. ANALOG TELEPHONY - 1 Baseband voice signal
Hz (CCITT, now called ITU-T)
Hz (Bell) We will use the ITU-T definition

12. ANALOG TELEPHONY - 2 KEY POINT
THE NUMBER OF VOICE CHANNELS A SATELLITE TRANSPONDER CAN CARRY VARIES INVERSELY WITH THE AVERAGE POWER LEVEL PER CHANNEL
Simple Example
NOTE:A pessimistic choice (power level set too high) will lower capacity estimate; An optimistic choice (power level set too low) can reduce quality of signals

13. CHANNEL LOADING EXAMPLE - 1
A 25 W transponder is designed to carry 250 two-way telephone channels (giving 500 channels at RF).
Q1. How much power is available for each telephone channel?
Answer: Power per channel = (25) / (500) = 50 mW
Q2. If the amplifier requires to be backed off 3 dB to preserve linearity, what is the power available per telephone channel now?
Answer: Power per channel = (25/2) / (500) = 25 mW
Q2. What is the power per channel in the second case if 1000 RF channels are carried?
Answer: Power per channel = (25 mW) / 2 = 12.5 mW

14. SATELLITE ANALOG Satellite transponders are bandwidth limited
A flexible scheme is therefore required for loading analog voice channels
earth stations may transmit in multiples of 12 voice channels (from 12 to 1872)
NOTE: There is very little analog (FM) voice traffic over satellites now. The bulk of the high capacity traffic is carried over optical fibers. The majority of voice capacity is in small digital carriers called IDR (Intermediate Digital Rate)

15. FREQUENCY MODULATION - 1
DEFINITION “Frequency modulation results when the deviation, f, of the instantaneous frequency, f, from the carrier frequency fc is directly proportional to the instantaneous amplitude of the modulating voltage”.
LET’S LOOK AT THIS PICTORIALLY

16. FREQUENCY MODULATION - 1
Input voltage
Transfer characteristic
Vmax
Instantaneous Input Voltage
Range of Input Voltage, v(t)
Instantaneous Output Frequency
Vmin f
Output Frequency
NOTE: In this example, fmin = the carrier frequency, fc
Range of Output Frequency
f min
f max

17. FREQUENCY MODULATION - 2
Schematic representation of a sinusoidal modulating signal, vp, on a carrier signal, frequency fc
NOTE: instantaneous frequency increases with increase in modulating voltage, and vice versa

18. FREQUENCY MODULATION - 3
The Frequency Modulated output signal, , will be as follows:
c = 2fc = carrier radian frequency
Maximum angular frequency deviation of the modulator
(5.2)
Maximum value of input modulating radian frequency

19. CARSON’S RULE - 1
Carson’s rule states that the transmission bandwidth, BT, is given by:
Where B is the bandwidth of the modulating signal which, for a sinusoidal modulating signal, is the highest modulating frequency, fmod.

20. CARSON’S RULE - 2
A. Single-frequency sinusoid: Approximate value for required bandwidth B:
(5.5)
Modulating frequency
Maximum frequency deviation
B. Real signal (practical case): Approximate value for required bandwidth B:
(5.6)
Maximum modulating frequency

21. FM IMPROVEMENT
FM modulation is relatively inefficient with the use of transmission spectrum
A small baseband bandwidth is converted into a large RF bandwidth
FM demodulation and detection converts the wide RF bandwidth occupied into a small baseband bandwidth occupied
Ratio of RF to baseband bandwidths gives an improvement in signal to noise ratio which leads to the so-called FM IMPROVEMENT

22. Digital Communications

23. DIGITAL COMMUNICATIONS -1
§5.4 in Chapter 5 + updated material
Many signals originate in digital form
data from computers
data from digital fixed and mobile systems
digitized information (e.g. voice)
World-wide network is moving towards all-digital system
Computers can only handle digital signals

24. Why Digital Transmission?
Robustness
Generally less susceptible to degradations
But...when it does degrade tends to fail quickly
Adaptiveness
Can easily combine a mix of signal information
Data, voice, video, multiple user signals
Compatibility - with digital storage, etc.
Security - not easily received except by recipient

25. DIGITAL COMMUNICATIONS -2
At baseband, send  V (volts) to represent a logical 1 and 0
At RF - digitally modulate the carrier
ASK Amplitude Shift Keying
FSK Frequency Shift Keying
PSK Phase Shift Keying
Binary forms of these are OOK, BFSK, and BPSK, respectively
Let’s first look at basic Digital Communications from the book by COUCH (7th. Edition)

26. DIGITAL COMMUNICATIONS -3
NOTE: from Couch

27. DIGITAL COMMUNICATIONS - 4
From Couch, Fig. 3-15

28. DIGITAL COMMUNICATIONS - 5
From Couch, Fig. 3-13

29. DIGITAL COMMUNICATIONS - 5
Analog-to-Digital recap; we have:
Sampled at 2 times highest frequency
Stored the sampled value
Compared stored value with a quantized level
Selected the nearest quantized level
Turned the selected quantized level into a digital value using the selected number of bits
We now need to generate a line code
Line Codes are serial bit streams that are used to drive the digital modulator

30. LINE CODES - 1 Couch Fig. 3-15 Usually used in digital circuits
Always have net zero voltage

31. LINE CODES - 2 SELECTION OF LINE CODE BASED ON
NEED TO HAVE SYNCHRONIZATION (OR OTHERWISE)
NEED TO HAVE A NET ZERO VOLTAGE (OR OTHERWISE)
NEED TO PREVENT STRING OF SAME VOLTAGE LEVEL SIGNALS
SPECTRAL EFFICIENCY
SOME TYPICAL SPECTRA

32. TYPICAL SPECTRA
Couch Fig. 2-6

33. PULSE SPECTRA
A random train of ones and zeroes has a spectrum (power spectral density) of
(5.40)
X = fTb, Tb = bit period, and f = frequency in Hz
Max value of Tb at f = 0
G(f) extends to f = 
Filtering affects the pulse shape

34. EFFECT OF FILTERING - 1
Fig. 5.8 in text

35. EFFECT OF FILTERING - 2
Rectangular pulses (i.e. infinite rise and fall times of the pulse edges) need an infinite bandwidth to retain the rectangular shape
Communications systems are always band-limited, so
send a SHAPED PULSE
Attempt to MATCH the filter to the spectrum of the energy transmitted
Before FILTERS, let’s look at Inter-Symbol Interference

36. INTER-SYMBOL INTERFERENCE
Sending pulses through a band-limited channel causes “smearing” of the pulse in time
“Smearing” causes the tail of one pulse to extend into the next (later) pulse period
Parts of two pulses existing in the same pulse period causes Inter-Symbol Interference (ISI)
ISI reduces the amplitude of the wanted pulse and reduces noise immunity
Example of ISI

37. ISI - contd. - 1
Form Couch, Fig. 3-23

38. ISI - contd. - 2
To avoid ISI, you can SHAPE the pulse so that there is zero energy in adjacent pulses
Use NRZ; pulse lasts the full bit period
Use Polar Signaling (+V & -V); average value is zero if equal number of 1’s and 0’s
Communications links are usually AC coupled so you should avoid a DC voltage component
Then use a NYQUIST filter
Nyquist Filter???

39. NYQUIST FILTER - 1 Bit Period is Tb
Sampling of the signal is usually at intervals of Tb
Thus, if we could generate pulses that are at a one-time maximum at t = Tb and zero at each succeeding interval of Tb (i.e. t = 2Tb, 3Tb, ….. , NTb then we would have no ISI
This is called a NYQUIST filter

40. Sampling instant is CRITICAL
NYQUIST FILTER - 2
Sampling instant is CRITICAL
Impulse at this point t
Tb Tb Tb Tb

41. NYQUIST FILTER - 3
NOTE: At each sampling interval, there is only one pulse contribution - the others being at zero level
Fig. 5.9 in text

42. NYQUIST FILTER - 4
Arranging to sample at EXACTLY the right instant is the “Zero ISI” technique, first proposed by Nyquist in 1928
Networks which produce the required time waveforms are called “Nyquist Filters”. None exist in practice, but you can get reasonably close

43. NYQUIST FILTER - 5 Noise into receiver must be held to a minimum
Place half of Nyquist filter at transmit end of link, half at receive end, so that the individual filter transfer function H(f) is given by Vr(f)NYQUIST = H(f)  H(f) Filter is a “Square Root Raised Cosine Filter”
H(f) matches pulse characteristic, hence it is called a “matched filter”
Matched Filter

44. MATCHED FILTER - 1 Roll-off factor =  = (f / f0 )
where f0 = 6 dB bandwidth
B = absolute bandwidth (here shown for  = 0.5) and
B = f f0
f1 = start of ‘roll-off’ of the filter characteristic
6 dB f1 f0 B
Fig in text

45. BUT how much bandwidth is required for a given transmission rate???
MATCHED FILTER - 2
A Raised Cosine Filter gives a Matched Filter response
The “Roll-Off Factor”, , determines bandwidth of Raised Cosine Low Pass Filter (LPF)
Gives zero ISI when the output is sampled at correct time, with sampling rate of Rb (i.e. at a sampling interval of Tb)
BUT how much bandwidth is required for a given transmission rate???

46. NOTE: It is the Symbol Rate that is key to bandwidth, not the Bit Rate
BANDWIDTH REQUIRED - 1
Bandwidth required depends on whether the signal is at BASEBAND or at PASSBAND
Bandwidth needed to send baseband digital signal using a Nyquist LPF is Bandwidth = (1/2)Rb(1 + )
Bandwidth needed to send passband digital signal using a Nyquist Bandpass filter is bandwidth = Rb(1 + )
NOTE: It is the Symbol Rate that is key to bandwidth, not the Bit Rate

47. BANDWIDTH REQUIRED - 2
SYMBOL RATE is the number of digital symbols sent per second
BIT RATE is the number of digital bits sent per second
Different modulation schemes will “pack” different numbers of Bits in a single Symbol
BPSK has 1 bit per symbol
QPSK has 2 bits per symbol

48. BANDWIDTH REQUIRED - 3
OCCUPIED BANDWIDTH, B, for a signal is given by B = Rs ( 1 +  ) where Rs is the symbol rate and  is the filter roll-off factor
NOISE BANDWIDTH, BN, for a channel will not be affected by the roll-off factor of filter. Thus BN = Rs

49. BANDWIDTH EXAMPLE - 1 GIVEN:
Bit rate 512 kbit/s
QPSK modulation
Filter roll-off, , is  = 0.3
FIND: Occupied Bandwidth, B, and Noise Bandwidth, BN
SOLUTION: Symbol Rate = Rs = (1/2)  (512  103) = 256  103
2 bits per symbol
Number of bits/s

50. BANDWIDTH EXAMPLE - 2
Occupied Bandwidth, B, is B = Rs (1 +  ) = 256  103 ( ) = kHz
Noise Bandwidth, BN, is BN = Rs = 256 kHz
Now what happens if you have FEC?
Example with FEC

51. BANDWIDTH EXAMPLE - 3 SAME Example, but 1/2-rate FEC is now used
SOLUTION Symbol Rate, Rs = (1/2)  (2)  (512  103) = 512  103 symbols/s Occupied Bandwidth, B, is B = Rs ( 1 +  ) = kHz
2 bits per symbol
1/2-rate FEC used
Number of bits/s

52. BANDWIDTH EXAMPLE - 3
Noise Bandwidth, BN, is BN = Rs = 512  103 = 512 kHz
Summary:
High Modulation Index  More Bandwidth Efficient
FEC (Block or Convolutional)  Increases bandwidth required

53. Digital Modulations

54. Digital Modulations
In digital communications, the modulating signal is a binary or M-ary data.
The carrier is usually a sinusoidal wave.
Change in Amplitude: Amplitude-Shift-Keying (ASK)
Change in Frequency: Frequency-Shift-Keying (FSK)
Change in Phase: Phase-Shift-Keying (PSK)
Hybrid changes (more than one parameter).
Ex. Phase and Amplitude change: Quadrature Amplitude Modulation (QAM)

55. Binary Modulations – Basic Types
These two have constant envelope (important for amplitude sensitive channels)

56. Coherent and Non-coherent Detection
Coherent Detection (most PSK, some FSK):
Exact replicas of the possible arriving signals are available at the receiver.
This means knowledge of the phase reference (phased-locked).
Detection by cross-correlating the received signal with each one of the replicas, and then making a decision based on comparisons with pre-selected thresholds.
Non-coherent Detection (some FSK, DPSK):
Knowledge of the carrier’s wave phase not required.
Less complexity.
Inferior error performance.

57. Design Trade-offs Primary resources: Design goals: Transmitted Power.
Channel Bandwidth.
Design goals:
Maximum data rate.
Minimum probability of symbol error.
Minimum transmitted power.
Minimum channel bandiwdth.
Maximum resistance to interfering signals.
Minimum circuit complexity.

58. Coherent Binary PSK (BPSK)
Two signals, one representing 0, the other 1.
Each of the two signals represents a single bit of information.
Each signal persists for a single bit period (T) and then may be replaced by either state.
Signal energy (ES) = Bit Energy (Eb), given by:
Therefore 

59. Orthonormal basis representation
Gram-Schmidt Orthogonalization: basis of signals that are both ortogornal between them and normalized to have unit energy.
Allows representation of M energy signals {si(t)} as linear combinations of N orthonormal basis functions, where N<=M.
Ex.: N=2

60. BPSK representation
Let’s consider the unidimensional base (N=1) where:
Let’s also rewrite the signal amplitudes as a function of their energy:

61. BPSK representation
Therefore, we can write the signals s1(t) and s2(t) in terms of 1(t):
Which can be graphically represented as:

62. BPSK Physical Implementation

63. Detection of BPSK Actual BPSK signal is received with noise
We assume AWGN in this class
Other noise properties are possible
AWGN is a good approximation
Other noise models are more complex
Constellation becomes a distribution because of noise variations to signal

64. Recall Gaussian Distribution
Area to the right of this line represents Probability (x>x0)
= mean
=standard deviation  x0 x
Approximation for large positive values of y 
Where:
Both Q(.) and erfc(.) functions are integrals widely tabled and available as functions in Excel and calculators

65. Calculating Error Probability
Noise Spectral Density = N0
Noise Variance:
BPSK error probability

66. Bit Error Rate (BER) for BPSK
BER is therefore given by
Approximation
valid for Eb/No
greater than ~4 dB
Eb/No
(dB) BER
0 0.08
2 0.04
8 2*10-4
10 4*10-6
Note that these calculations are for synchronous detection

67. Ambiguity Resolution
We haven’t discussed yet how to tell which signal state is a 1 and which a 0
Because of variations in the signal path, its impossible to tell a priori
Two common approaches resolutions:
Unique Word
Differential Encoding

68. Unique Word Ambiguity Resolution
A specific, known unique word is sent
The unique word is sent at a known time in the data
The correct signal state is chosen as 1 to correctly decode the unique word
Usually implemented with two detectors - the output of the correct one is simply used
Could lead to problems until a new UW is RX if a phase slip occurs
All bits after slip will be received wrong!

69. Differential Encoding Ambiguity Resolution
Data is not transmitted directly
Each bit is represented by:
0 => phase shift of p radians
1 => no phase shift
in the carrier
This results in ~ doubling the BER since any error will tend to corrupt 2 bits
BER is then
Valid for BER<~0.01

70. Coherent Quaternary PSK (QPSK)
Four signals are used to convey information
This leads to a constellation of: when shown as a phasor referenced to the signal phase, q
Each of the two states represents a two bits of information
Constant Modulus =>

71. QPSK Constellation Representation
In this case we use the following orthonormal basis:
Which gives, after application of some trigonometric identities, the following constellation representation:

72. QPSK Constellation

73. QPSK Waveform

74. QPSK Physical Implementation
Note that the QPSK
signal can be seen to
be two BPSK signals
in phase quadrature

75. Bit Error Rate (BER) for QPSK
The BER is still the probability of choosing the wrong signal state (symbol now)
Because the signal is Gray coded (00 is next to 01 and 10 for instance but not 11) the BER for QPSK is that for BPSK:
BER (after a lot of derivation) is given by:
Approximation
valid for Eb/No
greater than ~4 dB
Note that Eb is here, not Es!

76. Frequency Shift Keying
Two signals are used to convey information
In principle, the transmitted signal appears as 2 sinx/x functions at carrier frequencies
Each of the two states represents a single bit of information
Each state persists for a single bit period and then may be replaced either state
BER is: 2x BPSK BER for coherent for non-coherent
Constant Modulus =>

77. Frequency Shift Keying

78. Other Modulations (cont.)
M-ary PSK
PSK with 2n states where n>2
Incr. spectral eff. - (More bits per Hertz)
Degraded BER compared to BPSK or QPSK
QAM - Quadrature Amplitude Modulation
Not constant envelope
Allows higher spectral eff.

79. M-ary PSK

80. M-ary QAM

81. Other Modulations OQPSK MSK QPSK
One of the bit streams delayed by Tb/2
Same BER performance as QPSK
MSK
QPSK - also constant envelope, continuous phase FSK
1/2-cycle sine symbol rather than rectangular

82. Shannon Bound
1948 Shannon demonstrated that, with proper coding a channel capacity of
Required channel quality
for error free communications
=>we’re doing much worse

83. Modulation Schemes Error Performance

84. M-ary PSK Error Performance

85. Operation Point Comparison

86. Error Detection and Correction

87. Coding position on a transmission system

88. Error Protection Coding
Three types to discuss
Parity Bits (error detection only, really a subset of BC)
Block Coding (eg. Reed-Solomon)
Convolutional Coding (eg. Viterbi or Turbo)
All impose an overhead on channel
Additional information must be transmitted
This additional information is the redundant information of the error coding
Block codes develop less coding gain but are (much) easier to process (esp. at high data rates)
Often advantageous to use both together
Gain depends on BER - must be careful here
Coding ~ necessary for non-lin. ch.s (discuss BER flare)
Forward Error
Correction codes

89. Parity Bits The data is parsed into uniform k-bit words
7 bits is a common data length
An extra bit is added to this to make an k+1 bit transmission word
The value of the k+1th bit is determined by:
Even parity:
Odd parity:
Doesn’t correct errors just detects, and only an odd number of errors (discuss why)

90. Block Codes - 1 The data is parsed into uniform k-bit blocks
Coder adds n-k unique redundant bits
An n-bit block is transmitted
Coder is memoryless - only this block used
Transmitted data rate is then:
Redundant bits used to correct errors

91. Block Codes - 2
Hamming, Golay, BCH, Reed-Solomon, maximal-length are different types of block codes
Important for this class
Depending on amount of redundancy added, block codes may be used to detect only or to actually correct bit errors.
Block codes correct burst errors (ie. adjacent errors) as well as they do random errors.
Not as powerful as convolutional

92. Ciclic Codes (block codes)

93. Convolutional Codes - 1 Process as sliding window of data
Use constraint length of k (window length)
Transmit at rate of where r is rate
Fairly high coding gain
Turbo codes are even higher (but harder)
Do not handle burst errors well
Coding Gain (dB) for various Viterbi codes

94. Convolutional Codes - 2

95. Trellis Coding - 1

96. Trellis Coding - 2

97. Interleaving and Code on Code
Problem: Noise often happens in bursts
Can use interleaving - spreading adjacent bits of convolutional code over time to avoid having adjacent bits corrupted
But, we still have a quandary:
Block codes are robust against bursts
Convolutional codes provide more gain
Solution: use both inner convolutional and outer block codes to get both effects

98. Summary of Useful Formulas

99. Summary of Digital Communications -1
Legend of variables mentioned in this section:
M = modulation size. (Ex: 2, 4, 16, 64)
Bw = Bandwidth in Hertz
 = Roll-off factor (from 0 to 1)
Gc = Coding Gain (convert from dB to linear to use in formulas)
Ov = Channel Overhead (convert from % to fraction : 0 to1)
BER = Bit Error Rate

100. Summary of Digital Communications - 2
Bits per Symbol:
Symbol Rate [symbol/second]:
Gross Bit Rate [bps]:
Net Data Rate [bps]:

101. Summary of Digital Communications - 3
Required Eb/No (assuming no coding) [adimensional]:
(function of modulation scheme and required bit error rate – see table later)
Required Eb/No (using coding gain) [adimensional]:
Required C/N [adimensional]:

102. Summary of Digital Communications - 4
Required Signal Strength [Watts]:
Where k = Boltzman constant = 1.38e-23 J/Hz
TS = System Noise Temperature
T0 = ambient temperature (usually 290 K)
F = System Noise figure in linear scale (not in dB)

103. BER Calculation as a Function of Modulation Scheme and Eb/No Available
Equations given on next slide are used to calculate the bit error rate (BER) given the bit energy by spectral noise ratio (Eb/No) as input.
These functions are used in their direct form for the bit error rate calculations. Excel and some scientific calculators provide the solution for the “erfc” function.
The formulas provided can be inverted by numerical methods to obtain the Eb/No required as a function of the BER.
Also possible to draw the graphic and obtain the “inverse” by graphical inspection.

104. BER Calculation as a Function of Modulation Scheme and Eb/No Available - 2