2nd lecture of digital communications, Random signals,

1. Basic Communication Blocks
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2. Classification of signals
• Deterministic and random signals
– Deterministic signal: No uncertainty with respect to
the signal value at any time.
– A song played by your music player
– Random signal: Some degree of uncertainty in
signal values before it actually occurs.
• Thermal noise in electronic circuits due to the random
movement of electrons
• Number of cars passing on Peshawar road in front of the
college per hour
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3. Classification of signals
• Periodic and non-periodic signals
• Analog and discrete signals
A discretesignal
Analog signals
A non-periodic signalA periodic signal
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4. Classification of signals ..
• Energy and power signals
– A signal is an energy signal if, and only if, it has nonzero but
finite energy for all time and average zero power:
– A signal is a power signal if, and only if, it has finite but
nonzero power for all time:
– General rule: Periodic and random signals are powersignals.
Signals that are both deterministic and non-periodic are energy
signals.
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5. Autocorrelation
• Autocorrelation of an energy signal
• Autocorrelation of a power signal
– For a periodic signal:
• Autocorrelation of a random signal
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6. Spectral density
• Determinstic Energy signals:
– Energy spectral density (ESD):
• Determinstic Power signals:
– Power spectral density (PSD):
• Random process:
– Power spectral density (PSD):
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7. Random process
■ A random process is a function of a random variable evolving
with time.
■ For fixed value of the random variable: a random process is a
deterministic time signal.
■ For fixed t: a random process is a random variable
■ If one scans all possible outcomes of the underlying random
experiment, we shall get an ensemble of signals.
■ Random Process can be continuous or discrete
■ Random process is also called stochastic process
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8. Random process
• A random process is a collection of time functions, or
signals, corresponding to various outcomes of a
random experiment. For each outcome, there exists a
deterministic function, which is called a sample
function or a realization.
Random variables
Sample functions
or realizations
(deterministic
function)
time (t)
Realnumber
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9. Random process …
• Strictly stationary: If none of the statistics of the
random process are affected by a shift in the time
origin.
• Wide sense stationary (WSS): If the mean and
autocorrelation function do not change with a shift
in the origin time.
• Ergodic process: A random process is ergodic in
mean and autocorrelation, if
and
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10. CHAPTER 2
Formatting and Baseband Modulation
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11. Formatting
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12. Difference Between Baseband
and Bandpass Transmission
■ Baseband: When you transmit without a sinusoidal
■ Bandpass: When you need a sinusoidal as a carrier for
transmission
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13. Formatting and Baseband
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14. What is Formatting?
 Information can take either of the three forms:
1. Textual information
2. Analog signals
3. Digital data
 Before the signals are transmitted over a digital
communication channel, an information bearing signal
must be converted to digital symbols (Formatting).
 The resulting digital symbols are then represented by
baseband waveforms (Pulse Modulation or Line Coding).
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15. Block Diagram
Block diagram representing formatting and transmission of baseband signals.
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16. Textual Data (1)
American Standard Code for Information Interchange (ASCII) for encoding alphanumerics
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17. Textual Data (2)
Extended Binary Coded Decimal Interchange Information (EBCDIC) for encoding alphanumerics

18. Message and Symbol
 Textual message comprises a sequence of alphanumeric
characters.
 Example: Hello, how are you.
 Textual message is converted into a sequence of bits, i.e. bit
stream or baseband signal.
 Symbols are formed by a group of k bits from a finite symbol set
of M=2k such symbols.
 A system using a symbol set size of M is referred to as an M-ary
system.
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19. Message and Symbol:
Example
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20. Formatting Analog Information
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21. Periodic Sampling
 Typically, discrete-time signals are formed by periodically
sampling a continuous-time signal : x(n)=xa(nTs)
The sampling interval Ts is the sampling period, and
fs=1/Ts is the sampling frequency in samples per second.
 The sampling process:
fs=1/Ts
Sa(t)
xs(t)xa(t)
Convert impulses
into samples
x(n)
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22. Sampling Theorem :
 A bandlimited continuous-time signal, with highest frequency(bandwidth) B
Hz, can be uniquely recovered from its samples provided that the sampling
rate Fs  2B samples per second.
 The frequency Fs = 2B is called the Nyquist sampling frequency.
 If the signal is sampled at less than the Nyquist rate, then the aliasing
occurs.
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23.  

n
sa nTtts )()( 
)()()()()()()(   



 n
ssa
n
saaas nTtnTxnTttxtstxtx 
)(txa
0 Ts 2Ts 3Ts 4Ts 5Ts 6Ts 7Ts 8Ts 9Ts 10Ts 11Ts
0 Ts 2Ts 3Ts 4Ts 5Ts 6Ts 7Ts 8Ts 9Ts 10Ts 11Ts
0 Ts 2Ts 3Ts 4Ts 5Ts 6Ts 7Ts 8Ts 9Ts 10Ts 11Ts
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24. Illustration of Ideal Sampling
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25.  The Fourier transform of the continuous-time sampled signal is a
periodic function consisting of a superposition of shifted replicas of
, scaled by 1/Ts .
Bs  
)(sX

0 s
n=1n=0 n=2n=-1n=-2
s
1/Ts
)(sX

0 sf
n=1n=0 n=2n=-1n=-2
sf
1/Ts
)( fXa

0Bf Bf
B 2For s  Bff 2For s 
)( fXs
)( fXa
The overlap of the Fourier transform of each of the
terms of the sampled signal is called aliasing 44

26. Example
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27. Telephone companies digitize voice by assuming a
maximum frequency of 4000 Hz. The sampling rate
therefore is 8000 samples per second.
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