The document discusses Fourier analysis and Fourier transforms. It explains that Fourier analysis can be used to break down non-sinusoidal waveforms into sums of sinusoidal components. The Fourier transform represents this decomposition in the frequency domain. The document provides examples of applying Fourier analysis to common waveforms like square waves. It also summarizes properties of the Fourier transform like differentiation, time scaling, and convolution. Finally, it discusses various types of filters and how they are implemented using components like resistors, capacitors, and inductors.

1. Chapter Two: - Transmission of
Signals and Spectral Analysis:
part II
Fourier Series And Fourier Transform of signal

2. 1.2
2.2 Fourier Series
• One method used to determine the characteristics
and performance of a communication circuit or
system, specifically for non-sine wave approach, is
Fourier analysis.
• The Fourier theory states that a periodic non-
sinusoidal waveform can be broken down into
individual harmonically related sine wave or cosine
wave components.
• A square wave is one classic example of this
phenomenon.

3. According to Fourier analysis, any
composite signal is a combination of
simple sine waves with different
frequencies, amplitudes, and phases.
A single-frequency sine wave is not useful
in communication systems;
we need to send a composite signal, a
signal made of many simple sine waves.
Simple signal Composite signal

4. 1.4
2.2 Fourier Series
Basic Concepts
– Fourier analysis states that a square wave is made up of a
sine wave at the fundamental frequency of the square
wave plus an infinite number of odd harmonics.
– Fourier analysis allows us to determine not only sine-
wave components in a complex signal but also a signal’s
bandwidth.

5. 1.5
2.2 Fourier Series
Time Domain Versus Frequency Domain
– Analysis of variations of voltage, current, or power
with respect to time are expressed in the time
domain.
– A frequency domain plots amplitude variations with
respect to frequency.
– Fourier theory gives us a new and different way to
express and illustrate complex signals, that is, with
respect to frequency.

6. 1.6
2.2 Fourier Series
Figure 2-: The relationship between time and frequency domains.

7. 1.7
2.2 Fourier Series
Figure : Common non-sinusoidal waves and it’s Fourier equation.

8. 1.8
2.2 Fourier Series
Trigonometric form of Fourier series:
A sinusoidal signal, x(t)=A sin ⍵0t is a periodic signal
with period T=2𝜋/⍵0.
Also, the sum of two sinusoids is periodic provided that
their frequencies are integral multiples of a fundamental
frequency ⍵0.
 x1(t)=Asin4⍵0t and x2(t)=Asin3⍵0t , the sum of the two signals is
periodic because 4⍵0 and 3⍵0 are integral multiple of ⍵0

9. 1.9
2.2 Fourier Series
We can show that a signal x(t), a sum of sine and cosine
functions whose frequencies are integral multiples of ⍵o,
is a periodic signal. Consider a signal x(t), a sum of sine
and cosine function whose frequencies are integral
multiple of w0.
a0, a1, . . . , b1, b2, . . . are constants and w0 is the
fundamental frequency

10. 1.10
2.2 Fourier Series
Evaluation of Fourier coefficients of the trigonometric Fourier
series
 The constants a0, a1, a2,……ak and b0, b1, b2,……bk are called
Fourier coefficients. To evaluate a0, we shall integrate both
sides of the equation for x(t) over one period t0 to t0+T at an
arbitrary time t0 .Thus
OR
To evaluate an and bn, we can use the following results

11. 2.2 Fourier Series
•If the waveform of x(t) is odd then an=0; a0=0
•If the waveform of x(t) is even then bn=0


T
dt
t
kw
A
0
0 )
sin(
 T
t
kw
kw 0
0
0
)
cos(
1










 
)
0
cos(
)
cos(
1
0
0








 
 T
kw
kw
A
 
1
)
2
cos(
1
0








 
 
k
kw
0

Example: Integrating sin function

12. 1.12
2.2 Fourier Series
Example 1 Find the trigonometric Fourier series of the
given figure below
Figure shows a periodic rectangular waveform which is symmetrical to
the vertical axis. Obtain its F.S. representation

13. 1.13
2.2 Fourier Series

14. 1.14
2.2 Fourier Series
Example 2: Find the trigonometric Fourier series for the
periodic signal

15. 1.15
2.2 Fourier Series

16. 1.16
2.2 Fourier Series

17. 1.17
2.2 Fourier Series
Exercise 1:- Find the F.S.C. for the continuous-time periodic
signal
with fundamental freq. w=𝛑

18. 1.18
2.2 Fourier Series
Exercise 2:- Obtain the trigonometric Fourier series for the wave
form shown in below figure.

19. 1.19
2.2 The continuous time Fourier Transform

20. 1.20
2.2 the continuous time Fourier Transform

21. 1.21
3.2.1 properties of Fourier Transform
1) Differentiation
Hence if
then
Now

22. 1.22
3.2.1 cont’d
2. Time scaling

23. 1.23
2.2.1 cont’d
3) Convolution theorem
If two signals x(t) and y(t) are Fourier Transformable, and their convolution is also
Fourier Transformable, then the Fourier Transform of their convolution is the
product of their Fourier Transforms.
4. linearity

24. 1.24
3.2.1 cont’d
5. translation(Time shifting)
6. Modulation (Frequency-Domain Shifting)

25. 1.25
3.2.1 cont’d
7. Dilation (Time- and Frequency-Domain Scaling)

26. 1.26
2.2.1 cont’d

27. 1.27
3.2.1 cont’d
Example 3: Obtain the F.T. of the signal e−atu(t) and
plot its magnitude and phase spectrum.

28. 1.28
3.2.1 cont’d

29. 1.29
3.2.1 cont’d

30. 1.30
3.2.1 cont’d

31. 1.31
3.2.1 cont’d

32. 1.32
2-3: Filters and Filtering
A filter is a frequency-selective circuit.
Filters pass certain frequencies and reject others.
Passive filters are created using components such
as: resistors, capacitors, and inductors that do not
amplify.
Active filters use amplifying devices such as
transistors and operational amplifiers.

33. 1.33
2-3: Filters and Filtering
• There are five basic kinds of filter circuits:
 Low-pass filters only pass frequencies below a
critical (cutoff) frequency.
 High-pass filters only pass frequencies above the
cutoff frequency.
 Band pass filters pass frequencies over a narrow
range between lower and upper cutoff frequencies.
 Band-reject filters reject or stop frequencies over
a narrow range between lower and upper cutoff
frequencies.
 All-pass filters pass all frequencies over a
desired range but have a predictable phase shift
characteristic.

34. 1.34
2-3: Filters and Filtering
Filter Types (ideal)

35. Both coils and capacitors offer an opposition to alternating current flow known as
reactance, which is expressed in ohms. Like resistance, reactance is an opposition
that directly affects the amount of current in a circuit.

38. Series RLC circuit and Reactance versus
frequency

39. 1.39
2-3: Filters and Filtering
• The simplest form of low-pass filter is the RC circuit
• The circuit forms a simple voltage divider with one
frequency-sensitive component, in this case the
capacitor.
• At very low frequencies, the capacitor has very high
reactance compared to the resistance and therefore
the attenuation is minimum.
• As the frequency increases, the capacitive
reactance decreases. When the reactance becomes
smaller than the resistance, the attenuation
increases rapidly.
Low Pass Filter

40. 1.40
2-3: Filters and Filtering
RC Filters
– RC filters use combinations of resistors and
capacitors to achieve a desired frequency response.
– Most RC filters are of the low-pass or high-pass
type.
– Any low-pass or high-pass filter is effectively a
frequency-dependent voltage divider.
– An RC coupling circuit is a high-pass filter because
the ac input component is developed across the
resistor while dc voltage is blocked by a capacitor.

41. 1.41
2-3: Filters and Filtering
Figure 2 : RC low-pass filter. ( a) Circuit. (b) Low-pass filter.

42. 1.42
2-3: Filters and Filtering
Low Pass Filter (practical)
dB=20log (0.707vin/vin)= -3dB
 In fig. wc is defined as the (3 dB) frequency (cutoff frequency )
3dB is frequency at which the amplitude is (1/2)1/2 = 0.707 times
the maximum amplitude.
 3dB is frequency at which the power is 0.5 times the maximum
power.
 The passing band for low pass filter is w< wc.

43. 1.43
2-3: Filters and Filtering
RC Filters: High-Pass Filter
– A high-pass filter passes frequencies above the
cutoff frequency with little or no attenuation but
greatly attenuates those signals below the cutoff.
– The basic high-pass filter is a voltage divider with
the capacitor serving as the frequency-sensitive
component.
– A high-pass filter can be implemented with a coil
and a resistor.

44. 1.44
2-3: Filters and Filtering
Figure : (a) RC high-pass filter. (b) RL high-pass filter.

45. 1.45
2-3: Filters and Filtering
RC Filters: RC Notch Filter
– Notch filters, also called bandstop or band-
reject filters, attenuate a narrow range of
frequencies around a center point (frequency).
– A simple notch filter implemented with resistors
and capacitors is called a parallel-T or twin-T
filter.
– The center notch frequency is calculated:
fnotch =
1
2πRC

46. 1.46
2-3: Filters and Filtering
Band pass Filter
Simple band pass filters (a) Series (b) Parallel

47. 1.47
2-3: Filters and Filtering
Figure : RC notch filter.

48. 1.48
2-3: Filters and Filtering
LC Filters
– LC filters use combinations of inductors and capacitors to
achieve a desired frequency response.
– They are typically used with radio frequency (RF)
applications.
 Pass band is the frequency range over which the filter
passes signals.
 Stop band is the range of frequencies outside the
passband; that is, the range of frequencies that is
greatly attenuated by the filter.
 Attenuation is the amount by which undesired
frequencies in the stop band are reduced.