1. SIGNALS AND SIGNAL SPACE EENG 3810/CSCE 3020
C H A P T E R 2
SIGNALS AND SIGNAL SPACE
EENG 3810/CSCE 3020
Instructor: Oluwayomi Adamo

2. Signals and Systems What is a signal? What is a System?
A set of data or information e.g telephone or TV signal
What is a System?
An entity that processes a set of signals (inputs) to yield another set of signals (outputs)
Example: a system that estimates the position of a target based on information from a radar
A system could be hardware (electrical, mechanical or hydraulic) or software (algorithm)
How do we quantify a signal that varies with time?
How do you device a measure V for the size of human?
Assuming Cylinder with radius r:
Tv or telephone signal is a function of time
How do we derive a single number V as a measure of size – use width and height – product of width and height
For more precision, you can average the product over the entire length.

3. Signal Energy and Power
Measure of a signal g(t):
Could be area under a signal g(t)
Signal Energy Eg :
g(t) is squared to prevent the positive and negative areas from cancelling out
Signal energy must be finite
For a signal to be finite:
Signal amplitude -> 0 as |t| -> ∞ if not Eg will not converge

4. Signal Energy and Power
Figure 2.1 Examples of signals: (a) signal with finite energy;
(b) signal with finite power.
If Eg is not finite (infinite)
Average power Pg (mean squared) : Time average of energy (if it exists)
RMS (root mean square) value of g(t) =
Pg exists if g(t) is periodic or has statistical regularities

5. Signal Energy and Power
Average may not exist if the above condition is not satisfied e.g a ramp g(t) = t, energy or power does not exist
Measure of signal strength and size
Signal energy and power - inherent characteristic
Good indicator of signal quality
The signal to noise ratio (SNR) or ratio of the message signal and noise signal power
Standard unit of signal energy and power
Joule (J) and watt (W) respectively,
Logarithmic scales used to avoid zeros and decimal points
Signal with average power of P watts (10 log10P) dBw or (30+10.log10P) dbM

6. Determine the suitable measure for this signal?
This signal approaches 0 as |t|  , therefore use the energy equation.
This signal does not approach  0 as |t|   and it is a periodic wave, therefore use the power equation where g2 is replaced with t2.
What is the RMS value of this signal? 6

7. Determine the power and rms value of
Periodic signal with
Suitable measure of size is power
- First term on the right hand side equals C2/2
- Second term is zero –integral appearing in this term is area under a sinusoid.
Area is at most the area of half cycle – positive and negative portion cancels
A sinusoid of amplitude C has a power of C2/2 regardless of angular frequency

8. Determine the power and rms value
g (t) = C1 cos (1t + 1) + C2 cos (2t + 2) 1 ≠ 2
This signal is the sum of two sinusoid signals.
Therefore, use the power equation.
Therefore, Pg = (C12 / 2) + (C22 / 2)
This Can be generalized

9. What is the suitable measure for this signal?
g (t) = Dejt
The signal is complex and periodic. Therefore, use the power equation averaged over T0.
|ejt| = 1 so that |Dejt|2 = |D|2 and 9

10. Classification of Signals – Continuous and Discrete time Signal
Continuous time and discrete time signals
Analog and digital signals
Periodic and aperiodic signals
Deterministic and probabilistic signals
Continuous Time Signal: A signal that is specified for every value of time t .eg audio and video recordings
Discrete Time Signal: A signal that is specified only at discrete points of t=nT e,g quarterly gross domestic product (GDP) or stock market daily averages

11. Figure 2.3 (a) Continuous time and (b) discrete time signals.

12. Analog and Digital Signals
Is analog signal and continuous time signal the same? What of discrete time signal and digital signal?
What is Analog Signal?
A signal whose amplitude can have values in continuous range (values can take on infinite (uncountable) values
What is Digital Signal?
A signal whose amplitude can take only finite number of values.
For a signal to qualify for digital, the values don’t have to be restricted to two values.
A digital signal whose values can take on M values is an M-ary signal
Continuous and discrete time signal qualify a signal along the x-axis while Analog and digital signal qualify the signal in terms of the amplitude (y-axis)
Note: Analog signal is not necessarily CTS neither is digital DTS

13. Identify the signals above
Figure 2.4 Examples of signals: (a) analog and continuous time; (b) digital and continuous time;
(c) analog and discrete time; (d) digital and discrete time.
Identify the signals above

14. Periodic and Aperiodic signal
What is a Periodic Signal?
A signal is said to be periodic if there exists a positive constant T0
For all t. The smallest values of T0 that satisfy the equation (periodicity condition) above is the period of the signal g(t)
A periodic signal remains unchanged if time shifted by 1 period. Must start at -∞ and continue forever
What is an Aperiodic Signal?
A signal that is not periodic
Figure 2.5 Periodic signal of period T0.

15. Energy and Power Signal
Energy Signal: A signal with finite energy. Satisfies:
Power Signal: A signal with finite and non zero power (mean square value). Fulfills:
Power is the time average of the energy
Since the averaging is over a large interval, a signal with finite energy has zero power
A signal with finite power has infinite energy
Ramp signal has infinite power and are neither energy nor power signal. Not all power is periodic

16. Deterministic and Random Signal
Deterministic Signal: A signal whose physical description (mathematical or graphical) is known.
Random Signal: Signal known by only its probabilistic description such as mean value, mean squared value and distributions.
All message signals are random signals for it to convey any information.

17. Useful Signal Operators
Time Shifting, Time Scaling, and Time inversion
If T is positive, the shift is to the right (delayed)
If T is negative, the shift is to the left (advanced)

18. Time Scaling
Time Scaling: The compression or expansion of a signal in time
The signal in figure 2.7, g(t) is compressed in time by a factor of 2
Whatever happens in g(t) at some instant t will be happening at the instant t/2
If g(t) is compressed in time by a factor a>1, the resulting signal is:
If expanded

19. Figure 2.7 Time scaling a signal.

20. Time Inversion A special case of time inversion where a=-1
Whatever happens at some instant t also happens at the instant –t
The mirror image of g(t) about the vertical axis is g(-t).

21. Time Inversion Example
For the signal g(t) in (a) below, the sketch of g(-t) is shown in (b)

22. Unit Impulse Signal Unit impulse function (Dirac delta) A unit Impulse
Visualized as a tall, narrow rectangular pulse of unit area
Width ε is very small, height is a large value 1/ε
Unit impulse is represented with a spike.

23. Unit Impulse Signal
Multiplication of unit impulse by a function that is continuous at t = 0
Multiplication of a function with an impulse (an impulse located at t=T) ( must is defined at t=T)
Area under the product of a function with an impulse is equal to the value of that function at the instant where the unit impulse is located (Sampling or sifting property)

24. Unit Step Function u(t)
A signal that starts after t=0 is called a causal signal. A signal g(t) is causal if:
g(t) = t<0

25. Figure 2. 12 (a) Unit step function u(t)
Figure 2.12 (a) Unit step function u(t). (b) Causal exponential e−atu(t).

26. Signal Representation
Signals and Vectors
Signal Representation
As series of orthogonal functions (Fourier series)
Fourier series allows signal to be represented as points in a generalized vector space (signal space)
Information can therefore be viewed in geometrical context

27. Signal and Vectors
Any vector A in 3 dimensional space can be expressed as
A = A1a + A2b + A3c
a, b, c are vectors that do not lie in the same plane and are not collinear
A1, A2, and A3 are linearly independent
No one of the vectors can be expressed as a linear combination of the other 2
a, b, c is said to form a basis for a 3 dimensional vector space
To represent a time signal or function X(t) on a T interval (t0 to t0+T) consider a set of time function independent of x(t)

28. Signal and Vectors X(t) can expanded as
N coefficients Xn are independent of time and subscript xa is an approximation

29. Signals and Vectors Signal g can be written as N dimensional vector
g = [g(t1) g(t2) ………… g(tN)]
Continuous time signals are straightforward generalization of finite dimension vectors
In vector (dot or scalar), inner product of two real-valued vector g and x:
<g,x> = ||g||.||x||cosθ θ – angle between vector g and x
Length of a vector x:
||x||2 = <x.x> 29

30. Component of a Vector in terms of another vector.
Vector g in Figure 1 can be expressed in terms of vector x
g = cx + e
g  cx
e = g - cx (error vector)
Figure 2 shows infinite possibilities to express vector g in terms of vector x
g = c1x + e1 = c2 x + e2
Figure 1
Figure 2 30

31. Scalar or Dot Product of Two Vectors
 is the angle between vectors g and x.
The length of the component g along x is:
Multiplying both sides by |x| yields:
Where:
Therefore:
If g and x are Orthogonal (perpendicular):
Vectors g and x are defined to be Orthogonal if the dot product of the two vectors are zero. 31

32. Components and Orthogonality of Signals
Concepts of vector component and orthogonality can be extended to CTS
If signal g(t) is approximated by another signal x(t) as :
The optimum value of c that minimizes the energy of the error signal is:
We define real signals g(t) and x(t) to be orthogonal over the interval [t1, t2], if:
We define complex signals* x1(t) and x2(t) to be orthogonal over the interval [t1, t2]: 32

33. Example
For the square signal g(t) find the component in g(t) of the form sin t. In order words, approximate g(t) in terms of sin t so that the energy of the error signal is minimum
Sin 4pi = 0
Sin 2pi = 0
Sin pi = 0
Cos pi = -1
Cos 2pi = 1

34. Energy of The Sum of Orthogonal Signals
If vectors x and y are orthogonal and z = x + y, then:
If signals x(t) and y(t) are orthogonal over the interval [t1, t2] and if z(t) = x(t) + y(t), the energy is: =

35. Signal Comparison: Correlation Coefficient
Two vectors g and x are similar if g has a large component along x
Correlation Coefficient for real signals:
Correlation Coefficient for complex signals:
The magnitude of the Correlation Coefficient is never greater than unity (-1  Cn  1). If the two vectors are equal then Cn = 1. If the two vectors are equal but in opposite directions then Cn = -1. If the two vectors are orthogonal then Cn = 0. 35

36. Signal Comparison: Correlation Coefficient
Use Eq. (2.48) to compute Cn.
Cn =

37. Example

38. Application of Correlation
Signal processing in radar, sonar, digital communication, electronic warfare etc
In radar, if transmitted pulse is g(t), received radar return signal is :
Detection is possible if:
Target present
Target absent
α is target reflection and attenuation loss,
t0 – round trip propagation delay,
w(t) –noises and interferences
Target present
Target absent

39. Application of Correlation
Digital Communication
Detection of the presence of one or two known waveform in the presence of noise
Antipodal Scheme: Selecting one pulse to be the negative of the other pulse where cn will be -1
If noise is present in the received signal. Threshold detector is used to detect signal.
Large margins should be used to prevent detection error.
Antipodal has the highest performance in terms of guarding against channel noise and pulse distortion

40. Correlation Functions
Cross-correlation function of two complex signals g(t) and z(t):
Autocorrelation Function
Correlation of a signal with itself.
Measures the similarity of the signal g(t) with its own displaced version
Autocorrelation function of a real signal is:

41. Orthogonal Signal Set
Vector can be represented as a sum of orthogonal vectors
Orthogonality of a signal x1(t) x2(t) x3(t)…..xN(t) over time domain [t1, t2]:
If all signal energies En = 1, then the set is normalized and called orthonormal set
An orthogonal set can be normalized by dividing xn(t) by
If orthogonality is complete:

42. Orthogonal Signal Set A signal g(t) can be represented by
This is called generalized Fourier series of g(t) with respect to xn(t)
Energy of the sum of orthogonal signals is equal to the sum of their energies (sum of individual components)
This is called Parseval’s theorem

43. Exponential Fourier Series
Examples of orthogonal sets are trigonometric (sinusoid) functions, exponential (sinusoid) functions
The set of exponentials is orthogonal over any interval of duration
A signal g(t) can be expressed over an interval of T0 seconds as an exponential Fourier series:

44. Exponential Fourier Series
The compact trigonometric Fourier series of a periodic signal g(t) is given by
In exponential Fourier series where C0 = D0:
Exponential Fourier series will be used in this course because it is more compact, expression for derivation is also compact

45. Find the exponential Fourier series for the signal

47. Exponential Fourier Spectra
Coefficients Dn is plotted as a function of ω
If Dn is complex, two plots are required: real and imaginary parts of Dn or
Plot of amplitude (magnitude) and angle of Dn
To plot |Dn| versus ω and versus ω and Dn must be expressed in polar form