COE 341: Data

1
COE 341 Data Computer Communications
(T081)Dr. Marwan Abu-Amara

• Chapter 5
• Data Encoding

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Encoding and Modulation Techniques
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Digital Analog Signaling

• Digital signaling
• Data source g(t) encoded into digital signal x(t)
• g(t) may be analog (e.g. voice) or digital (e.g.
  file)
• x(t) dependent on coding technique, chosen to
  optimize use of transmission medium
• Conserve bandwidth or minimize errors
• Analog signaling
• Based on continuous constant frequency signal,
  carrier signal (i.e. A cos(2?fct?) or A
  sin(2?fct?))
• Carrier signal frequency chosen to be compatible
  with transmission medium
• Data transmitted by carrier signal modulation by
  manipulating A, fc, and/or?

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Analog Signaling

• Modulation
• Process of encoding source data onto a carrier
  signal with frequency fc
• Operation on one or more of three fundamental
  frequency-domain parameters amplitude,
  frequency, and phase
• Input signal m(t)
• Can be analog or digital
• Called modulating signal or baseband signal
• Modulated signal s(t) is result of modulating
  carrier signal called bandlimited or bandpass
  signal
• Location of bandwidth on spectrum related to
  carrier frequency fc

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Baseband vs. Bandpass Signals

• Baseband Signal
• Spectrum not centered around non zero frequency
• May have a DC component
• Bandpass Signal
• Does not have a DC component
• Finite bandwidth around or at fc

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Encoding and Modulation Remarks

• Encoding is simpler and less expensive than
  modulation
• Encoding into digital signals allows use of
  modern digital transmission and switching
  equipment
• Basis for Time Division Multiplexing (TDM)
• Modulation shifts baseband signals to a different
  region of the frequency spectrum
• Basis for Frequency Division Multiplexing (FDM)
• Unguided media and optical fibers can carry only
  analog signals

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Encoding Techniques

• Digital data, digital signal
• Simple and inexpensive equipment
• Analog data, digital signal
• Data needs to be converted to digital form
• Digital data, analog signal
• Take advantage of existing analog transmission
  media
• Analog data, analog signal
• Transmitted as baseband signal easily and cheaply

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Digital Data, Digital Signal

• Digital signal
• Sequence of discrete, discontinuous voltage
  pulses
• Each pulse is a signal element
• Binary data encoded into signal elements
• Unipolar signal
• All signal elements have same sign (e.g. 0 5 V,
  1 10 V ? DC content)
• Polar signal
• One logic state represented by positive voltage
  the other by negative voltage (e.g. 0 5 V, 1
  -5 V ? ideally Zero DC content)

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Digital Data, Digital Signal

• Mark and Space
• Binary 1 and Binary 0 respectively
• Duration or length of a bit (Tb)
• Time taken for transmitter to emit the bit
• Data rate, R ( 1/Tb)
• Rate of data transmission in bits per second
  (bps)
• Duration of a Signal Element (Ts)
• Minimum signal pulse duration
• Modulation (signaling) rate (1/Ts)
• Rate at which the signal level changes with time
• Measured in bauds signal elements per second

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Digital Data, Digital Signal

• Data rate 1/1ms
• 1 M bps
• Signaling Rate for NRZI 1/1ms
• 1 M bauds
• Signaling Rate for Manchester 1/0.5ms
• 2 M bauds

Tb
Ts
Ts
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Interpretation of the Received Signal
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Interpreting Digital Signal at Receiver

• Receiver needs to know
• Timing of bits - when they start and end
• Signal level
• Sampling comparison with a threshold value
• Factors affecting successful interpretation of
  signals signal to noise ratio, data rate,
  bandwidth
• Increase in data rate increases bit-error-rate
  (BER)
• Increase in SNR decreases BER
• Increase in bandwidth allows for increase in data
  rate

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Comparison of Encoding Schemes

• Encoding scheme
• Mapping from data bits to signal elements
• Signal Spectrum
• Lack of high frequencies reduces required
  bandwidth
• Lack of dc component allows ac coupling via
  transformer ? providing isolation reducing
  interference
• Transfer function of a channel is worse near the
  band edges
• ? Concentrate power in the middle of the
  bandwidth
• Clocking
• Synchronizing transmitter and receiver
• Sync mechanism based on signal
• can be built into signal encoding

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Comparison of Encoding Schemes

• Error detection
• Can be built into signal encoding
• Signal interference and noise immunity
• Some codes are better than others
• Cost and complexity
• Higher signal rate ( thus data rate) lead to
  higher costs
• Some codes require signal rate greater than data
  rate

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Encoding Schemes

• Nonreturn to Zero-Level (NRZ-L)
• Nonreturn to Zero Inverted (NRZI)
• Bipolar AMI (alternate mark inversion)
• Pseudoternary
• Manchester
• Differential Manchester

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Nonreturn to Zero-Level (NRZ-L)

• Two different voltages for 0 and 1 bits
• Voltage constant during bit interval
• no transition, no return to zero voltage
• e.g. Absence of voltage for zero, constant
  positive voltage for one
• More often, negative voltage for one value (1)
  and positive for the other (0)
• Used to generate or interpret digital data by
  terminals
• An example of absolute encoding
• Encoding data as a signal level

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Nonreturn to Zero Inverted (NRZI)

• Nonreturn to zero inverted on ones
• Constant voltage pulse for duration of bit
• Data encoded as presence or absence of signal
  transition at beginning of bit time
• Transition (low to high or high to low) denotes a
  binary 1
• No transition denotes binary 0
• An example of differential encoding
• Info to be transmitted represented as changes
  between successive signal elements

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NRZ
()ve
()ve
Transition Denotes one
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NRZ pros and cons

• Pros
• Easy to engineer
• Make good use of bandwidth
• Cons
• Large dc component
• Lack of synchronization capability
• No signal transitions for long strings of 0s or
  1s
• Used for magnetic recording
• Not often used for signal transmission

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Differential Encoding

• Data represented by signal transitions rather
  than signal levels
• Advantages
• With noise, signal transitions are detected more
  easily than signal levels
• In complex transmission layouts, it is easy to
  accidentally lose sense of polarity

RX

• Effect of swapping terminals on
• NRZ-L
• NRZI

_
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NRZ pros and cons
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Multilevel Binary

• Use more than two signaling levels
• Bipolar-AMI (Alternate Mark Inversion)
• zero represented by no line signal
• one represented by positive or negative pulse
• one pulses alternate in polarity
• No loss of sync if a long string of ones (zeros
  still a problem)
• No net dc component
• Lower bandwidth
• Easy error detection

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Pseudoternary

• One represented by absence of line signal
• Zero represented by alternating positive and
  negative
• No advantage or disadvantage over bipolar-AMI

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Bipolar-AMI and Pseudoternary
All Single Pulse Errors- Detected
Double Pulse Error- Undetected
Adding
Canceling
Double Pulse Error- Detected
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Trade Off for Multilevel Binary

• Not as efficient as NRZ
• Each signal element only represents one bit
• Date RateR1/TB
• In a 3 level system could represent log23 1.58
  bits
• Receiver must distinguish between three levels
  (A, -A, 0)
• Requires approximately 3dB more signal power for
  same probability of bit error

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Theoretical Bit Error Rate for Various Encoding
Schemes
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Biphase

• Manchester
• Transition in middle of each bit period
• Transition serves as clock and data
• High to low represents zero
• Low to high represents one
• Used by IEEE 802.3 (Standard for baseband coaxial
  cable twisted pair CSMA/CD bus LANs)
• Differential Manchester
• Mid bit transition is clocking only
• Transition at start of a bit period represents
  zero
• No transition at start of a bit period represents
  one
• Note this is a differential encoding scheme
• Used by IEEE 802.5 (Token ring LAN)

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Manchester Encoding
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Differential Manchester Encoding
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(No Transcript)
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Biphase Pros and Cons

• Pros
• Synchronization on mid bit transition (self
  clocking)
• No dc component
• Error detection
• Absence of expected transition
• Con
• At least one transition per bit time and possibly
  two
• Maximum modulation rate is twice NRZ
• Requires more bandwidth

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Modulation (Signaling) Rate

• Data rate (R)
• Bits per second, or bit rate
• 1/Tb, where Tb is bit duration
• Modulation rate (D)
• Rate at which signal elements generated
• Measured in Baud
• Modulation Rate D R k
• R data rate in bps
• M signaling levels used ? L bits/signal
  element log2 M
• k signal elements per bit signal trans./bit
  trans. 1/L
• In General, Modulation Rate D R k R/log2 M

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Modulation (Signaling) Rate
k1
k2
So, for Manchester ? D k?R 2/Tb
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Scrambling

• Use scrambling to replace sequences that would
  produce constant voltage
• Filling sequence
• Must produce enough transitions to sync
• Must be recognized by receiver and replace with
  original
• Same length as original
• Not be likely to be generated by noise
• No dc component
• No long sequences of zero level line signal
• No reduction in data rate
• Error detection capability

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B8ZS

• Bipolar With 8 Zeros Substitution
• Based on bipolar-AMI
• If octet of all zeros and last voltage pulse
  preceding was positive encode as 000-0-
• If octet of all zeros and last voltage pulse
  preceding was negative encode as 000-0-
• Causes two violations of AMI code
• Unlikely to occur as a result of noise
• Receiver detects and interprets as octet of all
  zeros

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HDB3

• High Density Bipolar 3 Zeros
• Based on bipolar-AMI
• String of four zeros replaced with one or two
  pulses

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B8ZS and HDB3
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Digital Data, Analog Signal

• Transmission of digital data through public
  telephone network
• Public telephone system
• 300Hz to 3400Hz
• Use modem (modulator-demodulator)
• Encoding techniques modify one of three
  characteristics of carrier signal
• Amplitude gt Amplitude shift keying (ASK)
• Frequency gt Frequency shift keying (FSK)
• Phase gt Phase shift keying (PSK)
• Resulting signal has a bandwidth centered on
  carrier frequency

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Digital Data, Analog Signal
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Amplitude Shift Keying (ASK)

• Binary values represented by different amplitudes
  of carrier
• Usually, one amplitude is zero
• i.e. presence and absence of carrier is used
• For a carrier signal resulting signal is

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Amplitude Shift Keying (ASK)

• Inefficient up to 1200bps on voice grade lines
• Used to transmit digital data over optical fiber

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Frequency Shift Keying (FSK)

• Binary values represented by two different
  frequencies near carrier frequency
• Resulting signal is
• f1 and f2 are offset from carrier frequency fc by
  equal but opposite amounts

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Frequency Shift Keying (FSK)

• Less susceptible to error than ASK
• Up to 1200bps on voice grade lines
• Used for high frequency (3 to 30 MHz) radio
  transmission
• Even higher frequencies on LANs using coaxial
  cables

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FSK
Df
fc
Df
f2
f1
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Frequency Shift Keying (FSK)

• Full-duplex transmission over voice grade line
• In one direction fc is 1170 Hz with f1 and f2
  given by 11701001270 Hz and 11701001070 Hz
• In other direction fc is 2125 Hz with f1 and f2
  given by 21251002225 Hz and 21251002025 Hz

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Phase Shift Keying (PSK)

• Binary PSK Phase of carrier signal is shifted to
  represent different values
• Phase shift of 180o
• Differential PSK Two-phase system with
  differential PSK
• Phase shift relative to previous bit transmitted
  rather than some constant reference signal
• Binary 0 represented by sending a signal burst of
  same phase as previous signal burst
• Binary 1 represented by sending a signal burst of
  opposite phase as previous signal burst

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BPSK
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Differential PSK (DPSK)

• Phase shifted relative to previous signal
  element rather than some reference signal

• 1 Reverse phase 0 Do not reverse phase
• (A form of differential encoding)
• Advantage
• - No need for a reference oscillator at RX to
  determine absolute phase

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Quadrature PSK (QPSK)

• More efficient Bandwidth use by each signal
  element representing more than one bit
• Shifts of ?/2 (90o)
• Resulting signal is
• Each signal element represents two bits

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Quadrature PSK (QPSK)
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Quadrature Amplitude Modulation (QAM)
Constellation

• An extension of the QPSK just described
• Combines both ASK and PSK
• For example, ASK with 2 levels and
• PSK with 4 levels give 4 x 2 i.e. 8-QAM
• Up to M256 is possible
• Large bandwidth savings
• But some susceptibility to
• noise
• QAM used on asymmetric
• digital subscriber line
• (ADSL) and some wireless
• systems

M8, L 3
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QAM

• One stream is ASK modulated on a carrier of
  frequency fc
• Other stream is ASK modulated on a carrier of
  frequency fc shifted by 90o
• The two modulated signals are combined together
  and transmitted
• Transmitted signal can be expressed as

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Multilevel PSK (MPSK)

• Can use more phase angles and even have more than
  one amplitude!
• For example, 9600 bps modems use 12
  phase angles, four of which have 2
  amplitudes
• Gives 16 different signal elements ? M
  16 and L log2 (16) 4 bits
• Every signal element carries 4 bits
  (Data sent 4 bits at a time)
• Baud rate is only 9600/4 2400 bauds (OK for a
  voice grade line)
• Complex signal encoding allows high data rates to
  be sent on voice grade lines having a limited
  bandwidth

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Data Rate Modulation Rate

• In general
• D modulation rate (signals per second or bauds)
• R data rate (bits per second)
• M number of different signal elements
• L number of bits per signal element
• With line signaling speed (i.e. D) of 2400 baud
• For NRZ-L, data rate is D 1/Tb
• For PSK, using L16 different combinations of
  amplitude and phase, data rate is 9600 bps, R
  4D
• For bi-phase, data rate is ½ D

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Performance of D/A Modulation Schemes

• Performance of digital-to-analog techniques
  depends on the definition of the bandwidth of the
  modulated signal
• Bandwidth of modulated signal depends on factors
  such as Filtering technique used to create the
  band-pass signal
• ASK and PSK bandwidth directly related to bit
  rate
• Transmission bandwidth BT for ASK and PSK is
• R is data rate
• r is related to filtering technique 0lt r lt1
• Transmission bandwidth BT for FSK is
• where the delta for offset from the carrier
  frequency

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Performance of D/A Modulation Schemes

• With multilevel signaling, bandwidth can improve
  significantly
• In the presence of noise, bit error rate of PSK
  and QPSK are about 3dB superior to ASK and FSK
  (as shown in Figure 5.4)

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Bandwidth Efficiency

• Bandwidth efficiency is the ratio of data rate to
  transmission bandwidth, R/BT

r 0 r 0.5 r 1
ASK 1.0 0.67 0.5
FSK (wideband ?F gtgt R) 0 0 0
FSK (narrowband ?F ? fc) 1.0 0.67 0.5
PSK 1.0 0.67 0.5
MPSK M4, L2 2.0 1.33 1.0
MPSK M8, L3 3.0 2.00 1.5
MPSK M16, L4 4.0 2.67 2.0
MPSK M32, L5 5.0 3.33 2.5
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Bandwidth Efficiency Bit Error Rate

• The bit error rate (BER) can be reduced by
  increasing Eb/N0
• Bit error rate can be reduced by decreasing
  bandwidth efficiency
• Increasing bandwidth
• Decreasing data rate
• N0 is the noise power density in watts/hertz.
  Hence, the noise in a signal with bandwidth BT,,
  NN0 BT

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Bandwidth Efficiency Bit Error Rate

• For multi-level signaling, replace R with D

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Example

• What is the bandwidth efficiency for FSK, ASK,
  PSK, and QPSK for a bit error rate of 10-7 on a
  channel with an SNR of 12dB ?
• Recall that Bandwidth efficiency is the ratio of
  R/BT

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Example

• For FSK ASK, Eb/N0 14.2dB
• ? (R/BT)dB 2.2 dB, R/BT 0.6
• For PSK, Eb/N0 11.2dB
• ? (R/BT)dB 0.8 dB, R/BT 1.2
• For QPSK, DR/2 (biphase) ? R/BT 2.4

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Analog vs. Digital Signaling Bandwidth Req.

• For digital signaling, bandwidth requirement is
  approximated to be
• For NRZ, D R

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Analog Data, Digital Signal

• Digitization
• Conversion of analog data into digital data
• Digital data can be transmitted using NRZ-L
• Digital data can be transmitted using code other
  than NRZ-L
• Digital data can then be converted to analog
  signal
• Analog to digital conversion done using a codec
• Pulse code modulation
• Delta modulation

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Pulse Code Modulation (PCM)

• Sampling Theorem If a signal is sampled at
  regular intervals at a rate higher than twice the
  highest signal frequency, the samples contain all
  the information of the original signal
• Signal maybe constructed from samples using a
  low- pass filter
• Voice data limited to below 4000Hz
• Require 8000 sample per second
• Analog samples (Pulse Amplitude Modulation, PAM)
• Each sample assigned digital value

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Quantization
Levels are numbered 0 to 15
n 4 bits ? 24 16 Quantization levels
PAM Sample
Analog signal is band-limited with bandwidth B
Quantization Error
Transmitted Serial Code representing the PAM
Samples
Sampling rate 2B
Each PAM sample is assigned the number of the
nearest quantization level and its digital code
is transmitted
Must finish sending the n bits of the code before
the next sample is due!
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Pulse Code Modulation (PCM)

• 4 bit system gives 16 levels
• Quantized
• Quantizing error or noise
• Approximations mean it is impossible to recover
  original exactly
• SNR for quantizing error is
• For each additional bit used for quantizing, SNR
  increases by about 6 dB or a factor of 4
• 8 bit sample gives 256 levels
• Quality comparable with analog transmission
• 8000 samples per second of 8 bits each gives
  (8000?8) 64 kbps

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PCM Example

• Suppose that we want to code an analog signal
  that has voltage levels 0-5v using 2-bit PCM
• Then, we divide the the voltage level in four
  intervals such that the size of each interval is
  5/41.25
• 0-1.25, 1.25-2.5, 2.5-3.75, 3.75-5
• We choose the values to be in the middle of each
  interval
• Selected values are 0.625, 1.875, 3.125, 4.375
• This guarantees that the maximum quantization
  error is ½5/40.625
• and quantization SNR 6 x 2 1.76 13.76 dB

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Nonlinear Encoding

• Absolute error for each sample is the same
  regardless of signal level
• Lower amplitude values are relatively more
  distorted
• Solution is to make quantization levels not
  evenly spaced
• Greater number of quantization steps for lower
  amplitudes and smaller number of steps for higher
  amplitudes
• Reduces overall signal distortion

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Effect of Nonlinear Coding
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Companding

• Effect of nonlinear coding can also be reduced by
  companding
• Compressing-expanding
• More gain to weak signals than to strong signals
  on input
• Reverse operation at output

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Example (Problem 5-20)

• Consider an audio signal with spectral components
  in the range of 300 to 3000 Hz. Assuming a
  sampling rate of 7000 samples per second will be
  used to generate the PCM signal.
• For SNR 30 dB, what is the number of uniform
  quantization levels needed?
• (SNR)dB 6.02 n 1.76 30 dB
• n (30 1.76)/6.02 4.69
• Rounded off, n 5 bits ? 25 32 quantization
  levels
• What data rate is required?
• R 7000 samples/sec ? 5 bits/sample 35 Kbps

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Delta Modulation

• Analog input is approximated by a staircase
  function
• Move up or down one level (?) at each sample
  interval
• Binary behavior
• Function moves up or down at each sample interval
• A bit stream produced approximates derivative of
  analog signal rather than its amplitude
• Produce a 1 if stair function is to go up
• Produce a 0 if stair function is to go down

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Delta Modulation - example
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Delta Modulation - Operation

• Analog input compared to most recent value of
  approximating staircase function
• If value exceeds staircase function, generate a 1
• Otherwise generate a 0
• Output of DM process is a binary sequence to be
  used for reconstructing staircase function
• Reconstructed stair function is smoothed by a low
  pass filter to reconstruct approximated analog
  signal

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Delta Modulation - Operation
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Delta Modulation

• Two important parameters in DM scheme
• Size of step assigned to each binary digit d
• Must be chosen to produce a balance between two
  types of errors or noise
• If waveform changes slowly, quantizing noise
  increases with increase in d
• If waveform changes rapidly, slope overload noise
  increases with decrease in d
• Increasing sampling rate
• improves the accuracy of the scheme
• Increases data rate
• Principal advantage of DM is implementation
  simplicity
• PCM has better SNR at same data rate

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CODEC - Performance

• Good voice reproduction
• PCM - 128 levels (7 bit)
• Voice bandwidth 4 KHZ
• Data rate should be 8000 x 7 56 kbps for PCM
• Bandwidth requirement
• Digital transmission requires 56 kbps for 4 KHz
  analog signal
• Using Nyquist theorem, this signal requires in
  the order of 28 KHz of Bandwidth, (C/256/2)

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CODEC - Performance

• A common PCM scheme for color TV uses 10-bit
  codes
• For bandwidth4.6 MHz ? 92 Mbps (i.e. 24.610)
• Digital techniques continue to grow in popularity
• Repeaters used with no additive noise
• Time-division multiplexing (TDM) is used for
  digital signals with no intermodulation noise
• Use more efficient digital switching techniques
• More efficient codes are used to reduce required
  bit rate

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Analog Data, Analog Signals

• Modulation
• Combining an input signal m(t) and a carrier at
  frequency fc to produce signal s(t) with
  bandwidth centered on fc
• Why modulate analog signals?
• Higher frequency may be needed for effective
  transmission
• For unguided transmission, impossible to send
  baseband signals as required antennas would be
  kilometers in diameter
• Permits frequency division multiplexing
• Types of modulation
• Amplitude
• Frequency
• Phase

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Analog Modulation
Amplitude Modulation (AM)
Angle Modulation (Phase, PM)
Angle Modulation (Frequency, FM)
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Amplitude Modulation

• Simplest form of modulation
• Accos 2pfct is the carrier,
• and x(t) Amcos 2pfmt is the input modulating
  signal
• Modulated signal expressed as
• na is the modulation index (? 1)
• Added 1 is a DC component to prevent loss of
  information there will always be a carrier
• Scheme is known as double sideband transmitted
  carrier (DSBTC)

Amplitude of modulated wave
Portion of the modulating signal
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Amplitude Modulation - Example

• Given the amplitude-modulating signal x(t)Amcos
  2pfmt , find s(t)
• Resulting signal has three components
• At the original carrier frequency fc
• A pair of additional components each
• spaced fm Hz from the carrier
• Envelope of resulting signal is 1na x(t)
• With na lt1, envelope is exact reproduction of the
  modulating signal,
• So it can be recovered at receiver
• With na gt1, envelope crosses the time axis and
  information is lost

Ac
Am/2
Am/2
fc
fm
fm
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Amplitude Modulation - Examples
MatLab Simulations
Modulating Signal
Carrier
Envelope
Modulated Signal
na 0.5/1 0.5
(10.5cos2pit) (1nacos2pit)
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Amplitude Modulation - Example
na 1/1 1
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Amplitude Modulation - Example
na 2/1 2 (gt1)
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Spectrum of an AM signal
Modulating Signal having a single Frequency, fm
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Spectrum of an AM signal
Modulating Signal having a finite Bandwidth, B

• Spectrum of AM signal is original
• carrier plus spectrum of original
• signal translated on both sides of fc
• Portion of spectrum f gt fc is
• upper sideband
• Portion of spectrum f lt fc is
• lower sideband
• Example voice signal 300-3000Hz
• With fc 60 KHz
• Upper sideband is 60.3-63 KHz
• Lower sideband is 57-59.7 KHz
• Bandwidth Requirement 2B

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Amplitude Modulation

• Total transmitted power Pt in modulated s(t) is
  given by
• Pc is transmitted power in carrier
• na should be maximized (but lt1) to allow most of
  signal power to carry information
• Modulated signal contains redundant information
• Only one of the sidebands is enough to restore
  modulating signal
• Possible ways to economize on transmitted power
• SSB single sideband, eliminates one sideband and
  carrier, saves on BW ( B)
• DSBSC double sideband suppressed carrier,
  carrier is not transmitted, no saving on BW (
  2B)
• Suppressing the carrier may not be OK in some
  applications, e.g. ASK, where the carrier can
  provide TX-RX synchronization.

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DSBSC Double Sideband Suppressed Carrier -
Example

• Signal is expressed as

Suppressed Carrier
90
Angle Modulation

• Includes
• Frequency modulation (FM) and
• Phase modulation (PM) as special cases
• Modulated signal is given by
• Phase modulation (PM)
• Instantaneous Phase is proportional to modulating
  signal
• np is phase modulation index
• Frequency modulation (FM)
• Instantaneous frequency deviation is proportional
  to modulating signal
• i.e. Derivative of f is proportional to
  modulating signal
• nf is frequency modulation index

Total Angle
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Angle Modulation

• The total phase angle of s(t) at any instant is
  2pfctf(t)
• Instantaneous phase deviation from carrier is
  f(t)
• Instantaneous angular frequency, , can be
  defined as the rate of change of total phase
• So, for the modulated signal, s(t)
• Phase Modulation (PM)
• f(t) npx(t), instantaneous phase deviation from
  carrier is directly proportional to x(t)
• Frequency Modulation (FM)
• f(t) is proportional to x(t). So, instantaneous
  frequency deviation from carrier frequency is
  proportional to x(t).

92
Phase Modulation (PM)- Example

• Derive an expression for a phase-modulated signal
  s(t) with Ac 5V, given the modulating signal
• x(t) 3 sin 2pfmt
• We know that s(t)
• For PM, f (t) is given by
• Then s(t) is
• Instantaneous frequency of s(t) is

Note Frequency variations in s(t) lead x(t)
amplitude variations by 90?
93
Frequency Modulation FM

• Peak frequency deviation DF is given by
• Where Am is the peak value of the modulating
  signal x(t)
• An increase in the amplitude Am of x(t) increases
  DF, which increases the bandwidth requirement BT
• But average power level of the FM modulated
  signal is fixed at AC2/2, (does not increase with
  Am)
• In Amplitude Modulation, Am affects the power in
  the AM signal, but does not affect the bandwidth

94
Frequency Modulation - Example

• Derive an expression for a frequency-modulated
  signal s(t) with Ac 5V, given the modulating
  signal
• x(t) 3 sin 2pfmt
• We know that s(t)
• For FM, f(t) is given by
• Then s(t) is
• We have
• Substituting for DF we get

But frequency varies as f, i.e. as sin not as
cos !!
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Bandwidth Requirement

• All AM, FM, and PM result in a modulated signal
  whose bandwidth is centered at fc
• Let B be the bandwidth of the modulating signal
• For AM, BT 2B
• Angle modulation includes a term of the form
  cos(cos()) which is a nonlinear term producing
  a wide range of frequencies fcfm, fc2fm,
  (the Bessel function)
• i.e. Theoretically, an infinite bandwidth is
  required to transmit an FM or PM signal

96
Practical Bandwidth Requirement for Angle
Modulation

• Carsons Rule of thumb
• For FM, BT 2DF 2B
• Both FM and PM require greater bandwidth than AM