Course: BEng (Hons) in Electrical and Electronic Engineering

1. EEEC6440315 COMMUNICATION SYSTEMS
Spectral Analysis
FACULTY OF ENGINEERING AND COMPUTER TECHNOLOGY
BENG (HONS) IN ELECTRICALAND ELECTRONIC ENGINEERING
Ravandran Muttiah BEng (Hons) MSc MIET

2. Fourier Transform
Direct Fourier Transform of 𝑔 𝑡 ,
𝐺 𝑓 =
−∞
+∞
𝑔 𝑡 e−j2π𝑓𝑡
d𝑡
Inverse Fourier Transform of 𝐺 𝑓 ,
𝑔 𝑡 =
−∞
+∞
𝐺 𝑓 ej2π𝑓𝑡
d𝑡
Symbolically;
𝐺 𝑓 = 𝐹 𝑔 𝑡
𝑔 𝑡 = 𝐹−1
𝐺 𝑓
We can plot 𝐺 𝑓 as a function of 𝑓, and amplitude and angle (phase) spectra exist,
𝐺 𝜔 = 𝐺 𝜔 ej𝜃g 𝜔
1

3. Find the Fourier Transform of e−𝑎𝑡
𝑢 𝑡 ,
𝐺 𝜔 =
−∞
∞
e−𝑎𝑡
𝑢 𝑡 e−j𝜔𝑡
d𝑡 =
0
∞
e− 𝑎+𝜔 𝑡
d𝑡 =e− 𝑎+𝜔 𝑡
d𝑡 =
−1
𝑎 + j𝜔
e− 𝑎+j𝜔 𝑡
0
∞
e−j𝜔𝑡
= 1
𝑡 → ∞, e− 𝑎+j𝜔 𝑡
= e−𝑎𝑡
e−j𝜔
= 0 if a > 0
𝐺 𝜔 =
1
𝑎+j𝜔
a > 0
Expressing 𝑎 + j𝜔 in polar form 𝑎2 + 𝜔2ejtan−1 𝜔
𝑎
𝐺 𝜔 =
1
𝑎2 + 𝜔2
e−jtan−1 𝜔
𝑎
𝐺 𝜔 =
1
𝑎2+𝜔2
and 𝜃g 𝜔 = −tan−1 𝜔
𝑎
2

4. Shifting The Phase Spectrum Of A Modulated Signal
By using cos 2π𝑓0𝑡 + 𝜃 instead of cos 2π𝑓0𝑡
If signal 𝑔 𝑡 is multiplied by cos 2π𝑓0𝑡 + 𝜃 ,
𝑔 𝑡 cos 2π𝑓0𝑡 + 𝜃0 ⇔
1
2
= 𝐺 𝑓 − 𝑓0 ej𝜃0 + 𝐺 𝑓 + 𝑓0 e−j𝜃0
If 𝜃0 = −
π
2
𝑔 𝑡 sin2π𝑓0𝑡 ⇔
1
2
𝐺 𝑓 − 𝑓0 e−j
π
2 + 𝐺 𝑓 + 𝑓0 ej
π
2
3

5. Example 1:
Find and sketch the Fourier transform of the modulated signal
𝑔 𝑡 cos 2π𝑓0𝑡 + 𝜃0 in which 𝑔 𝑡 is a rectangular pulse
𝑡
𝑇
as shown in
figure 1:
From pair,
𝑡
𝑇
⇔ sinc π𝑓𝑇
𝑔 𝑡 cos2π𝑓0𝑡 ⇔
1
2
𝐺 𝑓 − 𝑓0 + 𝐺 𝑓 + 𝑓0
4

6. 5
Figure 1: Modulated signal.
𝑔 𝑡 𝐺 𝑓
1
𝑡 𝑓
𝑡 𝑓
𝑇
2
−
𝑇
2 −
1
𝑇
1
𝑇
−
𝑇
2
𝑇
2
𝑔 𝑡 cos 𝜔0𝑡
𝑇
2
−𝑓0 𝑓0
𝐺 𝑓 − 𝑓0
2
𝐺 𝑓 + 𝑓0
2
0
2
𝑇
⇔
⇔

7. Application Of Modulation
Modulation is used to shift signal spectra in this scenarios:
• Interference will occur if signals occupying the same frequency band are
transmitted over the same medium.
• Example, if radio stations broadcast audio signals, receiver will not be able
to separate them.
• Radio stations could be assigned different carrier frequency.
• Radio station transmit modulated signal, thus shifting the signal spectrum to
allocated band.
• Both modulation and demodulation utilizes spectra shifting.
• Transmitting several signals simultaneously over a channel by using
different frequency bands is called Frequency Division Multiplexing.
• Audio frequencies are so low (large wavelength) that it will require
impractical large antennas for radiation.
• Shifting the spectrum to a higher frequency (smaller wavelength) by
modulation will solve the problem.
6

8. Signal Transmission Through A Linear System
Linear Time-Invariant (LTI) system can be characterised in the time domain or
frequency domain. The LTI system model can be used to characterise
communication channels. The stable LTI system can be characterised in the
time domain by its impulse response, ℎ 𝑡 . Impulse response is the system
response to a unit impulse input. The system response to a bounded signal 𝑥 𝑡
follows: 𝑦 𝑡 = ℎ 𝑡 ∗ 𝑥 𝑡 . Taking Fourier transform Y 𝑓 = 𝐻 𝑓 ∙ 𝑋 𝑓
(convolution theorem).
LTI System
ℎ 𝑡
𝐻 𝑓
Time domain 𝑥 𝑡
Frequency domain X 𝑓
Input Signal Output Signal
y 𝑡 = ℎ 𝑡 ∗ 𝑥 𝑡
Y 𝑓 = 𝐻 𝑓 ∙ 𝑋 𝑓
Figure 2: Linear time invariant system.
7

9. Analysis And Transmission Of Signal
Fourier transform of the impulse response ℎ 𝑡 , given as 𝐻 𝑓 , is
called transfer function or frequency response of LTI system.
𝐻 𝑓 is complex,
𝐻 𝑓 = 𝐻 𝑓 𝑒j𝜃h 𝑓
𝐻 𝑓 is the amplitude response.
𝜃h 𝑓 is the phase response of the LTI system.
8

10. Signal Distortion During Transmission
During transmission, signal 𝑥 𝑡 is changed to signal 𝑦 𝑡 . If X 𝑓 and Y 𝑓
are spectra of the input and output respectively, H 𝑓 is the spectra response,
Y 𝑓 = 𝐻 𝑓 ∙ 𝑋 𝑓
Output spectrum is given by the input spectrum multiplied by the spectral
response. Equation above expressed in polar form,
𝑌 𝑓 𝑒j𝜃y 𝑓
= 𝑋 𝑓 𝐻 𝑓 𝑒j 𝜃x 𝑓 +𝜃h 𝑓
Amplitude and phase relationship,
𝑌 𝑓 = 𝑋 𝑓 𝐻 𝑓
𝜃y 𝑓 = 𝜃x 𝑓 + 𝜃h 𝑓
During transmission, input signal amplitude 𝑋 𝑓 is changed to 𝑋 𝑓 ×
𝐻 𝑓 .
9

11. • Input signal phase spectrum 𝜃x 𝑓 is changed to 𝜃x 𝑓 + 𝜃h 𝑓 .
• An input signal is modified by a factor of 𝐻 𝑓 and shifted in phase by
𝜃h 𝑓 .
• Plot of 𝐻 𝑓 and 𝜃h 𝑓 as a function shows how the system is modifies the
amplitude and phase of various sinusoidal input.
• 𝐻 𝑓 is the frequency response of the system.
• During transmission some components are amplified and some are attenuated.
• Output will be different from input in general.
10

12. Distortion-Less Transmission
• In applications such as message transmission over communication channel,
the output waveform is required to be a replica of the input waveform.
• To achieve this, distortion due to amplification or communication channel
must be minimised.
• Distortion-less transmission is thus desired.
• Transmission is said to be distortion-less if the input and the output have
identical wave shapes within a multiplicative constant.
• A delayed output that retains the input waveform is distortion-less.
• Given input 𝑥 𝑡 and output y 𝑡 , a distortion-less transmission satisfies,
𝑦 𝑡 = 𝑘 ∙ 𝑥 𝑡 − 𝑡d
11

13. Fourier transform of previous equation:
𝑌 𝑓 = 𝑘𝑋 𝑓 e−j2π𝑓𝑡d
𝑌 𝑓 = 𝑋 𝑓 𝐻 𝑓
𝐻 𝑓 = 𝑘e−j2π𝑓𝑡d
𝐻 𝑓 = 𝑘
𝜃h 𝑓 = −2π𝑓𝑡d
For distortion-less transmission, amplitude response 𝐻 𝑓 must be a constant and
phase response 𝜃h 𝑓 must be linear function of 𝑓 going through the origin. The
slope of 𝜃h 𝑓 with respect to 𝜔 = 2π𝑓 is −𝑡d (delay of output with respect to
input).
Figure 3: Amplitude and phase response.
12
𝑘
0 𝑓
𝜃h 𝑓
𝐻 𝑓

14. Distortion Caused By Channel Non-Linearity
• The output signal contains new frequency components not present in the input
signal.
• In broadcast communication, signals are amplified at high power levels where
high efficiency amplifiers are desirable. However they (amplifiers) are non-
linear and they cause distortion. This is a problem in Amplitude Modulated
(AM) signals.
• Non-linear distortion does not affect Frequency Modulated (FM) signal.
• If a signal is transmitted over a non-linear channel, the non-linearity not only
distorts the signal, but also causes interference with other signals in the
channel because of spectral dispersion.
• Spectral distortion due to non-linear distortion causes interference among
signals using different frequency channels.
13

15. Example 2:
The input 𝑥 𝑡 and the output y 𝑡 of a certain nonlinear channel are related as,
𝑦 𝑡 = 𝑥 𝑡 + 0.000158𝑥2
𝑡 . Find the output signal, y 𝑡 and its spectrum,
Y 𝑓 if the input signal is, x 𝑡 = 2000sinc 2000π𝑡 . Verify that the bandwidth
of the output signal is twice that of the input signal due to signal squaring. Can the
signal 𝑥 𝑡 be recovered without distortion from the output y 𝑡 ?
x 𝑡 = 2000sinc 2000π𝑡 ⇔ 𝑋 𝑓 =
𝑓
2000
𝑦 𝑡 = 𝑥 𝑡 + 0.000158𝑥2
𝑡
= 2000sinc 2000π𝑡 + 0.316 × 2000sinc2
2000π𝑡
⇕
𝑌 𝑓 =
𝑓
2000
+ 0.316 ∆
𝑓
4000
14

16. • 0.316 × 2000sinc2
2000π𝑡 is the unwanted (i.e. distortion) term in
the received signal.
• The bandwidth of the received signal is twice that of the input signal
𝑥 𝑡 because of the squaring.
• The received signal contains the input signal 𝑥 𝑡 plus an unwanted
signal 632sinc2
2000π𝑡 .
• Spectra of the desired signal and the distortion overlap, it is impossible
to recover the signal 𝑥 𝑡 from the received signal y 𝑡 without
distortion.
• Passing through a low pass filter gives in the final figure, still some
residual distortion.
15

17. 16
𝑋 𝑓
1
0.316
1.316
1.316
−2000 2000
−2000 2000
−1000 1000
1000
−1000
−1000 1000 𝑓
𝑓
𝑓
𝑓
0
0
Figure 4: Input 𝑋 𝑓 and output 𝑌 𝑓 .
Distortion Term Spectrum
𝑌 𝑓

18. Signal Energy
Energy, 𝐸g of a signal 𝑔 𝑡 is defined as the area under 𝑔 𝑡 2
. We can determine the
energy signal from its Fourier transform G 𝑓 through Parseval’s theorem.
𝑔 𝑡 =
−∞
∞
𝐺 𝑓 𝑒j2π𝑓𝑡
d𝑡
𝐸g =
−∞
∞
𝑔 𝑡 𝑔∗
𝑡 d𝑡
=
−∞
∞
𝑔 𝑡
−∞
∞
𝐺∗
𝑓 e−j2π𝑓𝑡
d𝑓 d𝑡
=
−∞
∞
𝐺∗
𝑓
−∞
∞
𝑔 𝑡 e−j2π𝑓𝑡
d𝑡 d𝑓
=
−∞
∞
𝐺 𝑓 𝐺∗
𝑓 d𝑓
=
−∞
∞
𝐺 𝑓 2
d𝑓
Hence we can determine energy from both the time domain and the frequency domain.
17

19. Energy Spectral Density
The energy of a signal 𝑔 𝑡 is the result of energies contributed by all the
spectral components of the signal 𝑔 𝑡 . The contribution of a spectral
component of frequency 𝑓 is proportional to 𝐺 𝑓 . Assuming 𝑔 𝑡 is input
to a bandpass filter whose 𝐻 𝑓 is,
𝑌 𝑓 = 𝐺 𝑓 𝐻 𝑓
𝐸y =
−∞
∞
𝐺 𝑓 𝐻 𝑓 2
d𝑓
𝐸y = 2 𝐺 𝑓0
2
d𝑓
Energy spectral density (ESD) 𝛹g 𝑡 (per unit bandwidth in Hz) is defined as,
𝛹g 𝑓 = 𝐺 𝑓 2
𝐸g =
−∞
∞
𝛹g 𝑓 d𝑓
18

20. 19
The ESD of the signal 𝑔 𝑡 = e−𝑎𝑡
𝑢 𝑡
𝛹g 𝑓 = 𝐺 𝑓 2
=
1
2π𝑓 2 + 𝑎2
Figure 5: Spectral component of signal 𝐺 𝑓 .
−𝑓0 𝑓0
𝐻 𝑓
∆𝑓
1
𝑓
𝐺 𝑓0
2
−𝑓0 𝑓0 𝑓
𝑌 𝑓 2
𝐺 𝑓 2
0

21. Example 3:
Estimate the essential bandwidth, 𝑊 rad/s of the signal 𝑔 𝑡 = e−𝑎𝑡
𝑢 𝑡 if the
essential band is required to contain 95 % of the signal,
𝐺 𝑓 =
1
j2π𝑓 + 𝑎
𝐺 𝑓 2
=
1
2π𝑓 2 + 𝑎2
Signal energy 𝐸g is the area under ESD which 1
2𝑎
. If 𝑊 rad/s is the essential
bandwidth which contains 95 % of the total energy 𝐸g, 95 % of the area in the
figure,
0.95
2𝑎
=
−
𝑊
2π
𝑊
2π d𝑓
2π𝑓 2 + 𝑎2
=
1
2π𝑎
tan−1
2π𝑓
𝑎 𝑊
2π
𝑊
2π
=
1
π𝑎
tan−1
𝑊
𝑎
20

22. 21
Figure 6: Spectral of signal 𝐺 𝑓 .
𝐺 𝑓
−𝐵 𝐵
−𝑊 𝑊 𝜔
0
0.95π
2
= tan−1
𝑊
𝑎
⇒ 𝑊 = 12.7𝑎 rad/s
𝐵 =
𝑊
2π
= 2.02𝑎 Hz

23. (1) B. P. Lathi, Modern Digital and Analog Communication Systems, 4th Edition,
Oxford University Press, 2009.
(2) Leon W. Couch, Digital and Analog Communication Systems, 7th Edition,
Pearson, Prentice Hall, 2007.
(3) R. E. Ziemer and W. H. Tranter, Principles of Communications, Systems,
Modulation and Noise, 6th Edition, Wiley, 2009
References
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