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Wireless Information Transmission System Lab.

Chapter 4 Characterization of Communication Signals and Systems

Reference: Chapter 6, Communication Systems, 4th Edition. by Simon Haykin Institute of Communications Engineering National Sun Yat-sen University

Table of Contents
4.1 Representation of Band-Pass Signals and Systems
4.1.1 Representation of Band-Pass Signals 4.1.2 Representation of Linear Band-Pass Systems 4.1.3 Response of a Band-Pass System to a Band-Pass Signal 4.1.4 Representation of Band-Pass Stationary Stochastic Processes

4.2 Signal Space Representations
4.2.1 Vector Space Concepts 4.2.2 Signal Space Concepts 4.2.3 Orthogonal Expansions of Signals 2

Table of Contents
4.3 Representation of Digitally Modulated Signals
4.3.1 Memoryless Modulation Methods 4.3.2 Linear Modulation with Memory 4.3.3 Non-linear Modulation Methods with Memory – CPFSK and CPM

4.4 Spectral Characteristics of Digitally Modulated Signals
4.4.1 Power Spectra of Linearly Modulated Signals 4.4.2 Power Spectra of CPFSK and CPM Signals (*) 4.4.3 Power Spectra of Modulated Signals with Memory (*) 3

4.1 Representation of Band-Pass Signals and Systems The channel over which the signal is transmitted is limited in bandwidth to an interval of frequencies centered about the carrier. Signals and channels (systems) that satisfy the condition that their bandwidth is much smaller than the carrier frequency are termed narrowband band-pass signals and channels (systems). With no loss of generality and for mathematical convenience, it is desirable to reduce all band-pass signals and channels to equivalent low-pass signals and channels. 4

4.1.1 Representation of Band-Pass Signals
Suppose that a real-valued signal s(t) has a frequency content concentrated in a narrow band of frequencies in the vicinity of a frequency fc, as shown in the following figure:

Spectrum of a band-pass signal.

Our object is to develop a mathematical representation of such signals. 5

4.1.1 Representation of Band-Pass Signals
A signal that contains only the positive frequencies in s(t) may be expressed as: S + ( f ) = 2u ( f ) S ( f ) s+ ( t ) = ∫ S + ( f ) ⋅ e j 2π ft df
−∞ ∞

= F −1 ⎡ 2u ( f ) ⎤ ∗ F −1 ⎡ S ( f ) ⎤ ⎣ ⎦ ⎣ ⎦

where S( f ) is the Fourier transform of s(t) and u( f ) is the unit step function, and the signal s+(t) is called the analytic signal or the pre-envelope of s(t). j −1 F ⎡ 2u ( f ) ⎤ = δ ( t ) + ⎣ ⎦ πt j⎤ 1 ⎡ s+ ( t ) = ⎢δ ( t ) + ⎥ ∗ s ( t ) = s ( t ) + j ∗ s ( t ) ≡ s ( t ) + js ( t ) πt ⎦ πt ⎣ 6

4.1.1 Representation of Band-Pass Signals
Define:
1 1 s (t ) = ∗ s (t ) = πt π






s (τ ) t −τ

−∞



A filter, called a Hilbert transformer, is defined as: 1 −∞ 0 ) ⎪ ∞ 1 ∞ 1 − j 2π ft − j 2π ft H ( f ) = ∫ h (t ) e dt = ∫ e dt = ⎨0 ( f = 0) −∞ −∞ t π ⎪ ( f < 0) ⎩j 7

4.1.1 Representation of Band-Pass Signals
We observe that |H( f )|=1 and the phase response Θ( f )=-π/2 for f >0 and Θ( f )=π/2 for f 0) < 0)

⎧ 0 ⎪ then H ( − f − f c ) = ⎨ * ⎪H ( − f ) = H ( f ) ⎩ 14

(f (f

> 0) < 0)

4.1.2 Representation of Linear BandPass Systems
As a result H ( f ) = H l ( f − f c ) + H l* ( − f − f c ) thus h ( t ) = hl ( t ) e j 2π fc t + hl*e − j 2π fc t = 2 Re ⎡ hl ( t ) e j 2π fc t ⎤ ⎣ ⎦ where


=



-∞

H l* ( − f − f c ) e j 2π ft df using x = − f − f c ∞

= ∫ H l* ( x ) e − j 2π xt dx ⋅ e − j 2π fc t

(∫

−∞



−∞

Hl ( x) e

j 2π xt

dx ⋅ e − j 2π fc t = hl* ( t ) ⋅ e − j 2π fc t

)

*

In general, the impulse response hl(t) of the equivalent low-pass system is complex-valued. 15

4.1.3 Response of a Band-Pass System to a Band-Pass Signal
We have shown in Sections 4.1.1 and 4.1.2 that narrow band band-pass signals and systems can be represented by equivalent low-pass signals and systems. We demonstrate in this section that the output of a band-pass system to a band-pass input signal is simply obtained from the equivalent low-pass input signal...
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