Lecture 10 Slide 1PYKC 10-Feb-08E2.5 Signals & Linear SystemsLecture 10Fourier Transform (Lathi 7.1-7.3)Peter CheungDepartment of Electrical & Electronic EngineeringImperial College LondonURL: www.ee.imperial.ac.uk/pcheung/teaching/ee2_signalsE-mail: [email protected] 10 Slide 2PYKC 10-Feb-08E2.5 Signals & Linear SystemsDefinition of Fourier TransformX The forward and inverse Fourier Transform are defined for aperiodicsignal as:X Already covered in Year 1 Communication course (Lecture 5).X Fourier series is used for periodic signals.L7.1 p678Lecture 10 Slide 3PYKC 10-Feb-08E2.5 Signals & Linear SystemsConnection between Fourier Transform and Laplace TransformX Compare Fourier Transform:X With Laplace Transform:X Setting s = jωin this equation yield:X Is it true that: ?X Yes only if x(t) is absolutely integrable, i.e. has finite energy:L7.2-1 p697Lecture 10 Slide 4PYKC 10-Feb-08E2.5 Signals & Linear SystemsDefine three useful functionsX A unit rectangular window (also called a unit gate) function rect(x):X A unit triangle function Δ(x):X Interpolation function sinc(x):orL7.2-1 p687Lecture 10 Slide 5PYKC 10-Feb-08E2.5 Signals & Linear SystemsMore about sinc(x) functionX sinc(x) is an even function of x.X sinc(x) = 0 when sin(x) = 0 except when x=0, i.e. x = ±π,±2π,±3π…..X sinc(0) = 1 (derived with L’Hôpital’s rule)X sinc(x) is the product of an oscillating signal sin(x) and a monotonically decreasing function 1/x. Therefore it is a damping oscillation with period of 2π with amplitude decreasing as 1/x.L7.2 p688Lecture 10 Slide 6PYKC 10-Feb-08E2.5 Signals & Linear SystemsFourier Transform of x(t) = rect(t/τ)X Evaluation:X Since rect(t/τ) = 1 for -τ/2 < t < τ/2 and 0 otherwise ⇔L7.2 p689Bandwidth ≈ 2π/τLecture 10 Slide 7PYKC 10-Feb-08E2.5 Signals & Linear SystemsFourier Transform of unit impulse x(t) = δ(t)X Using the sampling property of the impulse, we get:X IMPORTANT – Unit impulse contains COMPONENT AT EVERY FREQUENCY.L7.2 p691Lecture 10 Slide 8PYKC 10-Feb-08E2.5 Signals & Linear SystemsInverse Fourier Transform of δ(ω)X Using the sampling property of the impulse, we get:X Spectrum of a constant (i.e. d.c.) signal x(t)=1 is an impulse 2πδ(ω).L7.2 p691orLecture 10 Slide 9PYKC 10-Feb-08E2.5 Signals & Linear SystemsInverse Fourier Transform of δ(ω - ω0)X Using the sampling property of the impulse, we get:X Spectrum of an everlasting exponential ejω0t is a single impulse at ω=ω0.L7.2 p692andorLecture 10 Slide 10PYKC 10-Feb-08E2.5 Signals & Linear SystemsFourier Transform of everlasting sinusoid cos ω0tX Remember Euler formula:X Use results from slide 9, we get:X Spectrum of cosine signal has two impulses at positive and negative frequencies.L7.2 p693Lecture 10 Slide 11PYKC 10-Feb-08E2.5 Signals & Linear SystemsFourier Transform of any periodic signalX Fourier series of a periodic signal x(t) with period T0is given by:X Take Fourier transform of both sides, we get:X This is rather obvious!L7.2 p693Lecture 10 Slide 12PYKC 10-Feb-08E2.5 Signals & Linear SystemsFourier Transform of a unit impulse trainX Consider an impulse train X The Fourier series of this impulse train can be shown to be:X Therefore using results from the last slide (slide 11), we get:L7.2 p69400() ( )TttnTδδ∞−∞=−∑0000021( ) where and jn tTn ntDe DTTωπδω∞−∞===∑Lecture 10 Slide 13PYKC 10-Feb-08E2.5 Signals & Linear SystemsFourier Transform Table (1)L7.3 p702Lecture 10 Slide 14PYKC 10-Feb-08E2.5 Signals & Linear SystemsFourier Transform Table (2)L7.3 p702Lecture 10 Slide 15PYKC 10-Feb-08E2.5 Signals & Linear SystemsFourier Transform Table (3)L7.3