DOC PREVIEW
SJSU EE 160 - Digital and Analog Communication Systems

This preview shows page 1-2-3 out of 10 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 10 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 10 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 10 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 10 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

EE160: Digital and Analog Communication SystemsSan Jose State UniversitySpring 2003Lecture Notes # 4February 10, 20031Real signals and the trigonometric Fourier seriesAs discussed in class:x(t)=a02+∞n=1ancos 2πntT0+ bnsin 2πntT0. (1)wherean=2T0α+T0αx(t)cos 2πntT0dt,bn=2T0α+T0αx(t)sin 2πntT0dt, (2)Since xn= x −n, it follows thatxnej2πntT0+ x−ne−j2πntT0=2|xn| cos 2πntT0+ ∠xn,and the following alternative Fourier trigonometric seriesexpansion of a real and periodic signal x(t) is obtainedx(t)=x0+2∞n=1|xn| cos 2πntT0+ ∠xn. (3)22.2 Fourier transfomThe Fourier transform is an extension of the Fourier seriesto other signals — periodic and nonperiodic.If x(t) is “well behaved” (meeting the Dirichlet condi-tions), then the Fourier transformof x(t), defined asX(f)=∞−∞x(t)e−j2πftdt, (4)exists.Notation: Capital letters denote signals in the frequency domain,lower-case letters denote signals in the time-domain.The original signal x(t) can be obtained from its Fouriertransform, via the inverse Fourier transform,x(t)=∞−∞X(f)ej2πftdf , (5)Notation• Operational: X(f)=F{x(t)},x(t)=F−1{X(f)}• Short-hand: x(t) ⇐⇒ X(f)3The Fourier transform of an impulseWrite a signal x(t) as the inverse Fourier transform of itsspectrumX(f), use the definition of X(f)intermsofx(t),and rearrange integrals:x(t)=∞−∞X(f)ej2πftdf=∞−∞ ∞−∞x(τ)e−j2πfτdτej2πftdf=∞−∞ ∞−∞ej2πf(t−τ)dfx(τ) dτ (6)From the definition of the unit impulse δ(t)asameasurefunction, a signal x(t) can be written asx(t)=∞−∞δ(t − τ)x(τ) dτ. (7)Then, combining Eqs. (6) and (7), we obtainδ(t − τ )=∞−∞ej2πf(t−τ)df , (8)or (redefining t as t − τ),δ(t)=∞−∞ej2πftdf . (9)This shows thatδ(t) ⇐⇒ 14Example: (Rectangular pulse)Find the Fourier transform of Π(t).Solution:F{Π(t)} =1/2−1/21 · e−j2πftdt=1−j2πfe−j2πft1/2−1/2=j2πf−ejπf+ e−jπf=−2j22πfsin(πf) = sinc(f).Π(t)tfsinc(f)11/2-1/211-123-3-252.1. Fourier transforms of real symmetric signalsx(t) X(f)real, even real, evenreal, odd imaginary, oddTo see this, write the Fourier transform using Euler’s for-mula for the complex exponential:X(f)=∞−∞x(t)e−j2πftdt=∞−∞x(t)cos(2πft) dt − j∞−∞x(t)sin(2πft) dt= XR(f) − jXI(f)If the signal x(t)isreal and even symmetric about zero,then x(t)sin(2πft) is real and odd symmetric and conse-quently, XI(f)=0.Similarly, if signal x(t)isreal and odd symmetric aboutzero,thenXR(f)=0.62.2.3 Fourier transform of periodic signalsLet x(t) be a periodic signal with period T0.Thenx(t)=∞n=−∞xnej2πnT0t⇐⇒ X(f)=∞n=−∞xnδ f −nT0,(10)where the following Fourier transform pair has been used(prove this as an exercise):ej2πnT0t⇐⇒ δ f −nT0.This shows that the Fourier transform of a periodic signalconsists of a sequence of impulses weighted by the Fourierseries coefficients xn.This suggests the following scheme for computing theFourier series of a periodic signal, based on properties ofthe Fourier transform:Use a “truncated signal” xT0(t) defined over a period ofx(t):xT0(t)=x(t), |t|≤T0/2;0, otherwise,(11)So that x(t) can be expressed asx(t)=∞n=−∞xT0(t − nT0) (12)7Observe that∞n=−∞xT0(t − nT0)=∞n=−∞xT0(t) δ(t − nT0)= xT0(t) ∞n=−∞δ(t − nT0) (13)Using the convolution theorem property and the Fouriertransform of a train of impulses:X(f)=XT0(f)1T0∞n=−∞δ f −nT0=1T0∞n=−∞XT0 nT0δ f −nT0(14)Comparing Eqs. (10) with (14), it is concluded thatxn=1T0XT0 nT0(15)This nice result gives a shortcut for computing the Fourierseries coefficients, based on the shape of a periodic signalover an interval (α, α + T0], ∀α.8Example: (Pulse train)xT0(t)=Π(t)tfsinc(f)11/2-1/211-123-3-2x(t)tnτ/T0sinc(nτ/T0)11/2-1/21/55-51015-15-10……Make periodicSam ple atMultiples of 1/T0(τ/T0= 1/5)T0-T09READING ASSIGNMENT• Review properties of the Fourier transform. Section2.2.2 of textbook.• Example 2.2.3 on page 40 of textbook.HOMEWORK # 1• To be assigned next class, 2/13/03• Due one week later,


View Full Document

SJSU EE 160 - Digital and Analog Communication Systems

Download Digital and Analog Communication Systems
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Digital and Analog Communication Systems and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Digital and Analog Communication Systems 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?