1. ELE 635 Chapter 3 Analysis and Transmission of Signals
Xavier Fernando
Spring 2015

2. Frequency and Time Domains
Certain aspects are better observed in the time domain using Oscilloscope.
Some other aspects are better observed in the frequency domain using Spectrum Analyser.
Fourier techniques provide tools to go back and forth.

3. Fourier Analysis
Periodic signals have a fundamental frequency and it harmonics.
Fourier series is used analyze periodic signals.
Fourier Transform is defined for Energy Signals
Discrete
Spectrum
Periodic
Waveform
Energy
Signal
Continuous
Spectrum

4. Fourier Analysis
Exponential Fourier Series is the most comprehensive way of looking at periodic signals.

5. Fourier Transform

6. Alternative definition

7. Fourier Series for Fourier Transform
Observe, 𝐷 𝑛 = 1 𝑇 𝑜 −𝑇 𝑜 /2 𝑇 𝑜 /2 𝑥(𝑡) 𝑒 −𝑗2𝜋𝑛 𝑓 𝑜 𝑡 𝑑𝑡
Also, 𝑋(𝑓)= −∞ ∞ 𝑥(𝑡) 𝑒 −𝑗2𝜋𝑓𝑡 𝑑𝑡
Hence, 𝑋(𝑓= 𝑛𝑓 𝑜 )= −∞ ∞ 𝑥(𝑡) 𝑒 −𝑗2𝜋𝑛 𝑓 𝑜 𝑡 𝑑𝑡
Comparing these two,
𝐷 𝑛 = 1 𝑇 𝑜 𝑋(𝑓= 𝑛𝑓 𝑜 )
*Integrating over − 𝑇 𝑜 /2 to 𝑇 𝑜 /2 of a periodic signal is same as integrating over -∞ to ∞ of one period of that signal

8. Fourier Series for Fourier Transform
Find the period 𝑇 𝑜 of the signal.
Take one period of the signal centered at f=0 (− 𝑇 𝑜 /2 to 𝑇 𝑜 /2).
Find the Fourier Transform X(f)
For n = 0, ±1, ±2, ±3,…find 𝐷 𝑛 = 1 𝑇 𝑜 𝑋(𝑓= 𝑛𝑓 𝑜 )
Then plot Dn
**Dn exists for only discrete values of f = nfo

9. Fourier Analysis

25. Real Frequency Shifting (Modulation) Properties

29. Spectrum of baseband Signals

30. In general a bandpass signal can be written as,
Slowly varying envelope
Slowly varying phase (freq.)

34. Bandwidth and Duration
The bandwidth and time-duration of a signal are inversely related and cannot be independently altered.

35. Basic Communication System
Channel – Linear Time Invariant (LTI) System
𝑦 𝑡 =ℎ 𝑡 ∗𝑥(𝑡)
𝑌 𝑓 =𝐻 𝑓 𝑋(𝑓)
𝑌 𝑠 =𝐻(𝑠)𝑋 𝑠
h(t): Impulse response of the LTI system
H(f): Frequency Response of the LTI system
H(s): Transfer function of the LTI system

39. Distortion Less Transmission

74. LTI System Response 𝑥 𝑡 =𝐴 sin (2𝜋 𝑓 𝑜 𝑡) HF(f)
Amplitude response multiplies.
Phase response is additive.