03 Signals Fourier-VP

8/3/2019 03 Signals Fourier-VP

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Signals, Fourier Series, Fourier Transform

Analog Digital Modulations

September 2011Prepared by

Assoc Prof. Thuong Le-Tien

Slides with references from HUT Finland, LaHore uni., Mc. Graw Hill Co., and A.B. Carlsons

Communication Systems book


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Review of Signals and Spectra A generic sinusoidal

signal

Phasor representation

Frequency domainrepresentation

Rotating phasors Frequency plots

Amplitude Phase

A

w0t +

f0f0 f

f0 f

A

Amplitude

Phase

000 2);cos()( ftAtv


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Two sided spectra can be seen from

This represents two rotating phasors

Amplitude and phase spectrum (two sided)


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Periodic Signals

A signal is periodic if there existssuch that

Smallest such is called fundamentalperiod

Any integer multiple of is also a period

x p(t)

x (t) = x (t + T)

T

T

T0

T0

T0


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Average signal and Power

Average signal

For periodic signals

Average power


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Fourier Series Representation Projection of periodic signals onto basis

functions

Periodic signal is a weighted sum of these basisfunctions

Exponentials are used as basis functions forwriting Fourier series

Any periodic signal can be expressed as asum of infinite number of exponentials (orsinusoids for real signals)


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Fourier Series

f0 = T01 (fundamental frequency)

Line spectra at frequencies that are integermultiple of fundamental frequency

DC component:


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Fourier series example:


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Fourier Series: Example


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Fourier Transform

Back to the Fourier series:


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Three major properties of V(f)


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Rectangular pulse spectrum

V() = A sinc Figure 2.2-2


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Signal transmission through linear systems

Deterministic signals:

Random signals:

Ideal distortion less transmission:

All the frequency components of the signal notonly arrive with an identical time delay, but alsoare amplified or attenuated equally.

Input Output

Linear system


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Ideal filters:

Realizable filters:

RC filters Butterworth filter

High-pass

Low-pass

Band-pass

Non-causal!


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Bandwidth of signal

Baseband versus bandpass:

Bandwidth dilemma:

Bandlimited signals are not realizable!

Realizable signals have infinite bandwidth!

Baseband

signal

Bandpass

signal

Local oscillator


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Bandwidth of signal

Different definition of bandwidth:a) Half-power bandwidth

b) Noise equivalent bandwidth

c) Null-to-null bandwidth

a) Fractional power containment bandwidth

b) Bounded power spectral density

c) Absolute bandwidth

(a)

(b)

(c)

(d)

(e)50dB

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03 Signals Fourier-VP