Chapter One

1
Chapter One

• Signals and Spectra

2
Why Digital ?

• Advantages
• Digital signals are more easily regenerated
• Digital circuits are more reliable and can be
  produced at lower cost
• Different types of digital signals can be treated
  as identical signals in transmission and
  switching
• Digital techniques are naturally to signal
  processing functions that protect against
  interference and jamming, or provide encryption
• Costs
• Very signal-processing intensive
• Need to synchronize at various levels
• Non-graceful degradation

3
Pulse Degradation and Regeneration
4
Typical Digital Communication System
5
Digital Communication Transformations

• Formatting
• Analog source audio, speech, video signal
• Digital source computer data, digital image
• Convert the source into a sequence of binary
  sequence
• Source encoding
• Efficiently convert the digital symbol into a
  sequence of binary digits
• Data compression MEG encode, JPEG, Huffiman
  coding, MP3
• Channel encoder
• Introduce some redundancy in the binary
  information sequence that can be used at the
  receiver to overcome the effects of noise and
  encounter the channel

6
Digital Communication Transformations

• Pulse modulation
• Map the binary information sequence into signal
  waveform
• Bandpass signaling
• Coherent PSK, FSK, GMSK
• Non-coherent DPSK, FSK

7
Basic Digital Communication Nomenclature
(Textual messages)
(Characters)
(7-bit ASCII)
(Symbol)
(Bandpass digital waveform)
8
Performance Criteria

• Analog communication systems
• The figure of merit is a fidelity criterion
• For example signal-to noise ratio, percent
  distortion, or expected mean-square error between
  the transmitted and received waveforms
• Digital communication systems
• Probability of incorrectly detecting a digit, or
  PE

9
Classification of Signals

• Deterministic and Random signals
• Deterministic signal means that there is no
  uncertainty with respect to its value at any
  time, for example x(t)5 cos 10t
• Random signal means that there is some degree of
  uncertainty before signal actually occurs
• Random waveform is NOT possible to write an
  explicit expression, can be described by
  probabilities and statistical averages
• Periodic and Non-periodic signals
• A signal x(t) is periodic in time if there exits
  a constant T0 such that
• No value of T0 that satisfies equation (1.2) is
  called non-periodic signal

10
Classification of Signals

• Analog and Discrete signals
• x(t) and x(kT)
• Energy and Power signals
• Energy signal is defined by the signal has
  nonzero but finite energy for all time
• Power signal is defined by the signal has finite
  but nonzero power for all the time
• Periodic signal and random signal are generally
  classified as power signals
• Both deterministic and non-periodic signals are
  generally classified as energy signals

11
Spectral Density

• Energy spectral density
• Where is defined as
  energy spectral density (ESD) of the signal x(t)
• Power spectral density
• The power spectral density (PSD) is
• See Example 1.1

12
Autocorrelation

• A measure of how closely the signal matches a
  copy of itself as the copy is shifted t in the
  time

13
Random Process
14
Random Process

• Stationary
• Strict-sense stationary if none of statistics are
  affected by a shift in the time origin
• Wide-sense stationary if
• Ergodic
• Time averages equal ensemble averages
• For example,
• The statistical properties of the process can be
  determined by time averaging over a single sample
  function

15
Some Useful Probability Distributions

• Binormial Distribution
• Let X be a discrete random variable X1 or X0,
  with probability p an 1-p
• Uniform Distribution
• Gaussian (normal) Distribution
• Chi-square (exponential) Distribution
• Rayleigh Distribution
• Ricean Distribution
• Lognormal Distribution

16
Autocorrelation and Power Spectral Density
17
Autocorrelation and Power Spectral Density
18
Normalized Gaussian Probability Density Function
19
White Noise
Figure 1.8 (a) Power spectral density of white
noise.(b) Autocorrelation function of white noise.
20
Linear Systems

• Frequency response
• Power spectral density
• Distortionless transmission

21
Ideal Filter

• Transfer function
• Impulse response

22
Impulse Response of the Ideal Low-pass Filter
23
Realizable Filter
24
Butterworth Filter

• Magnitude frequency response for the n-th order

25
RC Filtering an Ideal Pulse
26
Baseband versus Bandpass
27
Bandwidth Dilemma
Strictly bandlimited signal
Strictly time limited signal

• For all bandlimited spectra, the waveform are not
  realizable,
• and for all realizable waveforms, the absolute
  bandwidth is infinite.

28
Bandwidth Criteria
Fig. Bandwidth of digital data. (a) Half-power.
(b) Noise equivalent. (c) Null to null. (d) 99
of power. (e) Bounded PSD (defines attentuation
outside bandwidth) at 35 and 50 dB.