1. Lecture 2: SIGNALS
2nd semester
By: Elham Sunbu

2. Outline Signals and the classification of signals Sine wave
Time and frequency domains
Composite signals
Signal bandwidth
Digital signal
Signal operations

3. Introduction
A communication systems involves several stages of signal manipulation:
the transmitter transforms the message into a signal that can be sent over a communication channel;
the channel distorts the signal and adds noise to it;
and the receiver processes the noisy received signal to extract the message.
Thus, studying the communication systems must be based on a sound understanding of signals.

4. Signals
A signal is a physical quantity by which information can be conveyed; e.g. telephone and television signals.
Mathematically, a signal is represented as a function of an independent variable : time (t ) . Thus, a signal is denoted by s (t ), x (t ),… .
One way to show signals is by plotting them on a pair of perpendicular axes. The vertical axis represents the value or strength of a signal. The horizontal axis represents time.

5. Classification of Signals
There are several classes of signals:
Continuous-time and discrete-time signals.
Analog and digital signals.
Periodic and aperiodic signals.
Even and odd signals.

6. Continuous-time and Discrete-time Signals
Classification of Signals
Discrete-time signal:
is a signal that is specified only at discrete value of t.
( it is defined only at discrete values of t)
Continuous-time signal :
is a signal that is specified for every value of time t.
(it is defined for all time t)

7. Analog and digital signals
Classification of Signals
Analog signal:
is a signal whose amplitude can take on any value in a continues range.
Digital signal:
is a signal whose amplitude can take on only a finite number of values.

9. Periodic and Aperiodic Signals
A periodic signal completes a pattern within a measurable time frame, called a period, and repeats that pattern over subsequent identical periods.
The completion of one full pattern is called a cycle.
A signal is periodic with period T if there is a positive nonzero value of T for which
g(t+T) = g(t) for all t
An aperiodic (nonperiodic) signal changes without exhibiting a pattern or cycle that repeats over time.

10. T

11. Even and Odd Signals
An even signal is any signal f such that f (-t) = f (t). Even signals can be easily spotted as they are symmetric around the vertical axis.
An odd signal , on the other hand, is a signal f such that f (- t) = −f (t).

12. Simple and Composite Signals
Signals can be classified as simple or composite.
A simple signal is the signal that cannot be decomposed into simpler signals e.g. the sinusoidal signal (sine or cosine waves).
A composite signal is the signal that composed of multiple sinusoidal signals added together.

13. Sinusoidal Signals
Sinusoidal signals, based on sine and cosine functions, are the most important signals you will deal with. 
They are important because virtually every other signal can be thought of as being composed of many different sine and cosine signals.

14. Sine Wave A sine wave can be mathematically describe as
g(t) = A sin (ωt + φ)
where
A is the peak amplitude
ω is the angular frequency ω = 2πf f is frequency in Hertz ,
and φ is the phase.

15. Cosine Wave A cosine wave can be mathematically describe as
g(t) = A cos (ωt + φ)

16. Peak Amplitude
The peak amplitude of a signal is the absolute value of its highest intensity ( the largest value it takes), proportional to the energy it carries.
For electric signals, peak amplitude is normally measured in volts.

17. Period and Frequency
Period refers to the amount of time, in seconds, a signal needs to complete 1 cycle.
Frequency refers to the number of periods (cycles) in 1 s.
Note that period and frequency are just one characteristic defined in two ways.
Period is the inverse of frequency, and frequency is the inverse of period
and
second
Hertz

18. High frequency wave
Low frequency wave

19. Period and Frequency Period is formally expressed in seconds.
Frequency is formally expressed in Hertz (Hz), which is cycle per second.
Units of period and frequency are shown in the following table.

20. Examples
Q1) A sine wave has a frequency of 60 Hz, what is the period of this signal in ms ?
Solution:
F= 60 HZ , T= ? , so we use the equation of T:
T=1/f  1/60 (HZ) = s
so we convert from second to millisecond product not 10 −3 because from Second (big) to millisecond (small).
= X 10 3
=16.6 ms
Q2) Express a period of 100 ms in microseconds?
Convert from millisecond to second
100 ms X 10 −3 =0.1 s
so we convert from millisecond to second product 10 −3 not because from millisecond (small) to second (big).
2. Then product the result of second by microsecond
0.1 X 10 −6 = µs

21. Examples
Q3) The period of a signal is 100 ms . What is its frequency in kilohertz?
Solution:
T= 100 ms , f= ? , the first we convert from millisecond to second :
100 ms X 10 −3 =0.1 s
Then we use the equation of f:
f=1/T  1/0.1 = 10 HZ
* so we convert from Hertz to Kilohertz product not 10 −3 because from Hertz (small) to Kilohertz (big).
= 10 X 10 3
=10000 Kilohertz

22. More About Frequency
We already know that frequency is the relationship of a signal to time and that the frequency of a wave is the number of cycles it completes in 1 s.
But another way to look at frequency is as a measurement of the rate of change with respect to time.
Change in a short span of time means high frequency.
Change over a long span of time means low frequency.

24. More About Frequency
What if a signal does not change at all? What if it maintains a constant voltage level for the entire time it is active?  In such a case, its frequency is zero.
What if a signal changes instantaneously? What if it jumps from one level to another in no time?  Then its frequency is infinite.

25. Phase
The term phase describes the position of the waveform relative to time 0.
If we think of the wave as something that can be shifted backward or forward along the time axis, phase describes the amount of that shift. It indicates the status of the first cycle.

26. note
that the signal does not really exist before time 0.

27. Phase Phase is measured in degrees or radians.
A phase shift of 360° corresponds to a shift of a complete period.
A phase shift of 180° corresponds to a shift of one-half of a period.
A phase shift of 90° corresponds to a shift of one-quarter of a period.

28. Wavelength
Wavelength is another characteristic of a signal traveling through a transmission medium.
Wavelength depends on both the frequency and the medium.
The wavelength is the distance a simple signal can travel in one period.
The wavelength is normally measured in micrometers (microns) instead of meters.

29. Wavelength
Wavelength can be calculated if one is given the propagation speed (the speed of light) and the period (or frequency) of the signal
where λ is the wavelength and c is the propagation speed.

30. Wavelength

31. Time and Frequency Domains
The time-domain plot shows changes in signal amplitude with respect to time (it is an amplitude-versus- time plot).
To show the relationship between amplitude and frequency, we can use what is called a frequency-domain plot.
A frequency-domain plot is concerned with only the peak value and the frequency.
Changes of amplitude during one period are not shown.
It is obvious that the frequency domain is easy to plot and conveys the information that one can find in a time domain plot.
The advantage of the frequency domain is that we can immediately see the values of the frequency and peak amplitude.

32. A complete sine wave is represented by one spike.
The position of the spike shows the frequency and its height shows the peak amplitude.

33. The time domain and frequency domain of three sine waves
The frequency domain is more compact and useful when we are dealing with more than one sine wave.
The time domain and frequency domain of three sine waves

34. Composite Signals
A single-frequency sine wave is not useful in data communications
We need to send a composite signal, a signal made of many simple sine waves.
According to Fourier analysis, any composite signal is a combination of simple sinusoidal signals with different frequencies, amplitudes, and phases.
A composite signal can be periodic or non-periodic.
A periodic composite signal can be decomposed into a series of simple sinusoidal signals with discrete frequencies (that have integer values 1, 2, 3, and so on) in the frequency domain. (Fourier series)
A non-periodic composite signal can be decomposed into a combination of an infinite number of simple sinusoidal signals with continuous frequencies in the frequency domain. (Fourier transform)

35. A composite periodic signal
Example 1:

36. Example 2:
A non-periodic composite signal
Frequency domain

37. Signal Bandwidth
The range of frequencies contained in a composite signal is its bandwidth.
The bandwidth is normally a difference between two numbers.
Bandwidth of a periodic signal
Bandwidth of a non-periodic signal

38. Examples
Q1) If a periodic signal is decomposed into five sine waves with frequencies of 100, 300, 500, 700, and 900 Hz, what is its bandwidth? Draw the spectrum, assuming all components have a maximum amplitude of 10 V.
Solution
Let fh be the highest frequency, fl the lowest frequency, and B the bandwidth. Then
The spectrum has only five spikes, at 100, 300, 500, 700, and 900 Hz

39. Examples
Q2) A periodic signal has a bandwidth of 20 Hz. The highest frequency is 60 Hz. What is the lowest frequency?
Solution
Let fh be the highest frequency, fl the lowest frequency, and B the bandwidth. Then
The spectrum contains all integer frequencies. We show this by a series of spikes

40. Digital Signals Information can be represented by a digital signal.
Amplitude
A digital signal with two levels
Time
Amplitude
A digital signal with four levels
Time

41. Digital Signals
The bit interval is the time required to send one single bit.
The bit rate is the number of bits sent in 1 s, expressed in bits per second (bps).
1 s
1 s
Bit interval
Bit rate = 8 bps
Bit rate = 16 bps

42. Example
Q1) A digital signal has eight levels. How many bits are needed per level? We calculate the number of bits from the formula
Solution
Each signal level is represented by 3 bits.

43. Digital Signal as a Composite Analog Signal
It should be know that a digital signal with all its sudden changes is actually a composite signal having an infinite number of frequencies. In other word, the bandwidth of a digital signal is infinite.
Fourier analysis can be used to decompose a digital signal.

44. Some Signal Operations
We discuss here three useful signal operations: shifting, scaling, and inversion.
Since the independent variable in our signal description is time, these operations are discussed as time shifting, time scaling, and time reversal (inversion).
However, this discussion is valid for functions having independent variables other than time (e.g., frequency).

45. Time Shifting Shifts the signal left or right
Let x(t) denote a continues time signal.
If y(t) = x(t – a)  y(t) is a time shifted signal of x(t) by a seconds,
If a > 0  y(t) is a delayed version of x(t) (i.e. shift x(t) relative to the time axis towards the right by a )
If a < 0  y(t) is an advanced version of x(t) shift to left (i.e. shift x(t) relative to the time axis towards the left by a )

48. Time Scaling The compression or expansion of a signal in time.
Let  x(t) denote a continues time signal.
If y(t) = x(at)  y(t) is a time scaling version of x(t).
|a| > 1  compression x(t) by a factor of “a”
|a| < 1  expansion x(t) by a factor of “a”
Also, we can look to scaling as speed up or slow down a signal
|a| > 1  speed up x(t) by a factor of “a”
|a| < 1  slow down x(t) by a factor of “a”

51. Time Reversal ( Inversion)
Reflecting the signal about t=0.
Let  x(t) denote a continues time signal.
If y(t) = x(-t)  y(t) is a time reversal version of x(t).

53. Thank you