Introduction to Digital Communication Systems

ELFT 443 – Communications II Introduction to Digital Communication Systems

Digital and Analog Sources

Digital Source

Analog Source

Digital and Analog Communication System

Digital Communicaation System

Analog Communication System

Advantages of Digital Communication Systems

Disadvantages of Digital Communication Systems

Primary concerns of a Digital Communication System Designers 1. Selection of the information-bearing waveform 2. Bandwidth and power of the waveform 3. Effect of system noise on the received information 4. Cost of the system

- 1 – ermtiong ELFT 443 – Communications II Signals and Spectra

Conditions to Satisfy for Physically Realizable Waveforms: 1. The waveform has significant nonzero values over a composite time interval that is finite. 2. The spectrum of the waveform has significant values over a composite frequency interval that is finite. 3. The waveform is a continuous function of time. 4. The waveform has a finite peak values. 5. The waveform has only real values. At any time, it cannot have a complex value a + jb where b is nonzero.

Signals  An event that serves, or at least is capable, to start some action

Classification of Signals

I. Energy Signals and Power Signals

Energy Signals  usually exist for only a finite interval of time or even if present for an infinite amount of time, at least has a major portion of its energy concentrated in a finite time interval

Power Signals

II. Periodic and Non-Periodic Signals Periodic Signal  one that repeats itself exactly after a fixed length of time

- 2 – ermtiong ELFT 443 – Communications II

Non Periodic or Aperiodic  any signal for which there is no value of T satisfying the equation

- 3 – ermtiong ELFT 443 – Communications II III. Random and Deterministic Signals Random Signal  one about which there is some degree of uncertainty before it actually occurs  cannot be completely specified as a function of time and must be modelled probabilistically

Non Random or Deterministic Signal  can be modelled as a completely specified function of time

System  A group of objects that can interact harmoniously and that are combined in a manner intended to achieve a certain objective.

Classification of Systems

I. Linear and Non-Linear Systems

II. Time-Invariant and Time Varying Systems

III. Realizable and Nonrealizable

Time Average Operator

Definition:

 this is a linear operator, since the average of the sum of the two quantities is the same as the sum of their averages

- 4 – ermtiong ELFT 443 – Communications II Theorem: If the waveform involved is periodic, the time average operator can be reduced to

DC Value The DC value of a waveform w(t) is given by its time average

RMS value The root mean square value of w(t) is given by

- 5 – ermtiong ELFT 443 – Communications II Singularity Functions

Singularity functions have simple mathematical forms but are not finite everywhere or they do not have finite derivations of all orders everywhere.

I. Unit Impulse Function or Dirac Delta Function The dirac delta function is defined by

  w(x) xdx  w0  Where w(x) is any function that is continuous at x = 0.

Properties: 1. Sifting Property  w(x) x  x dx  w x   0   0  

The impulse function sifts out a particular value w(x0)from the integral.

2. Area (Strength)   xdx  1   x has a unit area. In an analogous manner, A x has an area of A units.

3. Amplitude

 t  t0   0 for all t  t 0 4. Graphic Representation

To avoid any attempt to display the amplitude at t = t 0, we shall

draw an arrow at the point t = t 0 as indicator of the impulse function. The area of the impulse is designated by a quantity in parenthesis beside the arrow.

5. Time Scaling A scaling in the argument of the impulse function can be converted as follows: - 6 – ermtiong ELFT 443 – Communications II 1  ax   x a 6. Symmetry  t    t  t is defined as an even function.

7. Multiplication by a Time Function

w(x) x  x0   wx0  x  x0 

w(x) is continuous at x 0

8. Equivalent Integral for the Dirac Delta Function   x   e j2xy dy 

II. Unit Step Function The unit step function u(t) is

III. Derivative of Singularity Functions From:   tdt  1  ut 

- 7 – ermtiong ELFT 443 – Communications II Fourier Transform and Spectra

Fourier Transform Definition:  Wf   wtej2ftdt 

W(f) is a two – sided spectrum of w(t) because both positive and negative frequency components are obtained from the equation.

Techniques for determining the Fourier Transform 1. Direct integration 2. Table of Fourier transforms 3. Fourier transform theorems 4. Superposition to break the problem into 2 or more simple problems 5. Differentiation or integration of w(t) 6. Numerical integration of the FT integral on the PC via MATLAB or MathCAD 7. Fast Fourier Transform (FFT) on the PC via MATLAB or MathCAD

Since ej2ft is complex, W(f) is a complex function of frequency, W(f) maybe decomposed into:

quadrature form: W(f) = X(f) +jY(f)

  polar form: Wf  Wfe j f

where: Wf  X 2 f Y2 f

1 Yf and f  tan   Xf

The time waveform maybe calculated from the spectrum by using the inverse Fourier transform:  wt   Wfe j2ftdf  Properties of Fourier Transforms: 1. Spectra symmetry of real signals W(-f) = W*(f)

2. Magnitude spectrum is even about the origin

- 8 – ermtiong ELFT 443 – Communications II W(f)  W(f) Phase spectrum is odd about the origin

 f  f

Parseval’s Theorem   * *  w1 tw2 (t)   W1 fW2 f  

Rayleigh’s Energy Theorem

  E   wt 2 dt   Wf 2 dt  

Energy Spectral Density

E  Wf 2

where: w(t) and W(f) is a Fourier Transform pair

Fourier Transform Theorems

Operation Function Fourier Transform

Linearity a1w1 t a2 w2 t a1W1 f a2 W2 f -jwTd Time delay w(t – Td) W1(f)e 1  f  Scale change w(at) W  a  a  Conjugation w*(t) W*(-f) Duality W(t) w(-f) Real signal frequency 1    w(t) cos (ω t+θ) e j Wf  f  e j Wf  f  translation c 2 c c [w(t) is real] Complex signal w(t)ejwct W(f – f ) frequency c translation

- 9 – ermtiong ELFT 443 – Communications II Bandpass 1 Re{g(t)ejwct} Gf  f  G *  f  f  signal 2 c c dn w(t) Differentiation j2fn Wf dtn t  1 Integration  wd j2f 1 W(f)  W0f  2 

Convolution w1(t) * w2(t)   w1 w2 t  d W1(f)W2(f) - 

Multiplication w1(t)w2(t) W1 f* W2 f   W1 W2 f  d  Multiplication dn Wf tnw(t)  j2fn by tn df n

- 10 – ermtiong