6.003: Signals and SystemsCT Fourier TransformApril 8, 2010CT Fourier TransformRepresenting signals by their frequency content.X(jω)=Z∞−∞x(t)e−jωtdt (“analysis” equation)x(t)=12πZ∞−∞X(jω)ejωtdω (“synthesis” equation)• generalizes Fourier series to represent aperiodic signals.• equals Laplace transform X(s)|s=ωif ROC includes jω axis.→ inherits properties of Laplace transform.• complex-valued function of real domain ω.• simple ”inverse” relation→ more general than table-lookup method for inverse Laplace.→ “duality.”• filtering.• applications in physics.FilteringNotion of a filter.LTI systems• cannot create new frequencies.• can only scale magnitudes and shift phases of existing components.Example: Low-Pass Filtering with an RC circuit+−vi+vo−RCLowpass FilterCalculate the frequency response of an RC circuit.+−vi+vo−RCKVL: vi(t) = Ri(t) + vo(t)C: i(t) = C ˙vo(t)Solving: vi(t) = RC ˙vo(t) + vo(t)Vi(s) = (1 + sRC)Vo(s)H(s) =Vo(s)Vi(s)=11 + sRC0.010.110.01 0.1 1 10 100ω1/RC|H(jω)|0−π20.01 0.1 1 10 100ω1/RC∠H(jω)|Lowpass FilteringLet the input be a square wave.t12−120Tx(t) =Xk odd1jπkejω0kt; ω0=2πT0.010.110.01 0.1 1 10 100ω1/RC|X(jω)|0−π20.01 0.1 1 10 100ω1/RC∠X(jω)|Lowpass FilteringLow frequency square wave: ω0<< 1/RC.t12−120Tx(t) =Xk odd1jπkejω0kt; ω0=2πT0.010.110.01 0.1 1 10 100ω1/RC|H(jω)|0−π20.01 0.1 1 10 100ω1/RC∠H(jω)|Lowpass FilteringHigher frequency square wave: ω0< 1/RC.t12−120Tx(t) =Xk odd1jπkejω0kt; ω0=2πT0.010.110.01 0.1 1 10 100ω1/RC|H(jω)|0−π20.01 0.1 1 10 100ω1/RC∠H(jω)|Lowpass FilteringStill higher frequency square wave: ω0= 1/RC.t12−120Tx(t) =Xk odd1jπkejω0kt; ω0=2πT0.010.110.01 0.1 1 10 100ω1/RC|H(jω)|0−π20.01 0.1 1 10 100ω1/RC∠H(jω)|Lowpass FilteringHigh frequency square wave: ω0> 1/RC.t12−120Tx(t) =Xk odd1jπkejω0kt; ω0=2πT0.010.110.01 0.1 1 10 100ω1/RC|H(jω)|0−π20.01 0.1 1 10 100ω1/RC∠H(jω)|Source-Filter Model of Speech ProductionVibrations of the vocal cords are “filtered” by the mouth and nasalcavities to generate speech.buzz fromvocal cordsspeechthroat andnasal cavitiesFilteringLTI systems “filter” signals based on their frequency content.Fourier transforms represent signals as sums of complex exponen-tials.x(t) =12πZ∞−∞X(jω)ejωtdωComplex exponentials are eigenfunctions of LTI systems.ejωt→ H(jω)ejωtLTI systems “filter” signals by adjusting the amplitudes and phasesof each frequency component.x(t) =12πZ∞−∞X(jω)ejωtdω → y(t) =12πZ∞−∞H(jω)X(jω)ejωtdωFilteringSystems can be designed to selectively pass certain frequency bands.Examples: low-pass filter (LPF) and high-pass filter (HPF).0ωLPF HPFLPFHPFtttFiltering Example: ElectrocardiogramAn electrocardiogram is a record of electrical potentials that aregenerated by the heart and measured on the surface of the chest.0 10 20 304050 60−1012t [s]x(t) [mV]ECG and analysis by T. F. WeissFiltering Example: ElectrocardiogramIn addition to picking up electrical responses of the heart, electrodeson the skin also pick up a variety of other electrical signals that weregard as “noise.”We wish to design a filter to eliminate the noise.filterx(t) y(t)0 10 20 304050 60−1012t [s]x(t) [mV]0 10 20 304050 60−1012t [s]y(t) [mV]Filtering Example: ElectrocardiogramWe can identify the “noise” by breaking the electrocardiogram intofrequency components using the Fourier transform.0.01 0.1 1 10 10010001001010.10.010.0010.0001f =ω2π[Hz]|X(jω)| [µV]low-freq.noisecardiacsignalhigh-freq.noise60 HzFiltering Example: ElectrocardiogramFilter design: low-pass flter + high-pass filter + notch.0.01 0.1 1 10 10010.10.010.001f =ω2π[Hz]|H(jω)|Electrocardiogram: Check YourselfWhich poles and zeros are associated with• the high-pass filter?• the low-pass filter?• the notch filter?s-plane( )( )( )222Electrocardiogram: Check YourselfWhich poles and zeros are associated with• the high-pass filter?• the low-pass filter?• the notch filter?s-plane( )( )( )222high-passlow-passnotchnotchFiltering Example: ElectrocardiogramBy placing the poles of the notch filter very close to the zeros, thewidth of the notch can be made quite small.59 60 6110.50f =ω2π[Hz]|H(jω)|Filtering Example: ElectrocardiogramComparision of filtered and unfiltered electrocardiograms.0 10 20 304050 60−1012t [s]x(t) [mV]0 10 20 304050 60−1012t [s]y(t) [mV]0.01 0.1 1 10 10010001001010.10.010.0010.0001f =ω2π[Hz]|X(jω)| [µV]low-freq.noisecardiacsignalhigh-freq.noise60 Hz0.01 0.1 1 10 10010001001010.10.010.0010.0001f =ω2π[Hz]|Y (jω)| [µV]Filtering Example: ElectrocardiogramReducing the frequency components that are not generated by theheart simplifies the output, making it easier to diagnose cardiacproblems.Unfiltered ECG0 10 20 304050 60012t [s]x(t) [mV ]Filtered ECG0 10 20 304050 6001t [s]y(t) [mV ]Continuous-Time Fourier Transform: SummaryFourier transforms represent signals by their frequency content.→ useful for many signals, e.g., electrocardiogram.→ motivates representing a system as a filter.→ useful for many systems.Visualizing the Fourier TransformFourier transforms provide alternate views of signals.11tPulses contain all frequencies except harmonics of 2π/width.↔2πω21tWider pulses contain more low frequencies than narrow pulses.↔4πω1tConstants (in time) contain only frequencies at ω = 0.↔2πωFourier Transforms in Physics: DiffractionA diffraction grating breaks a laser beam input into multiple beams.Demonstration.Fourier Transforms in Physics: DiffractionThe grating has a periodic structure (period = D).θλDsin θ =λDThe “far field” image is formed by interference of scattered light.Viewed from angle θ, the scatterers are separated by D sin θ.If this distance is an integer number of wavelengths λ → constructiveinterference.Fourier Transforms in Physics: DiffractionCD demonstration.Check YourselfCD demonstration.laser pointerλ = 500 nmCDscreen3 feet1 feetWhat is the spacing of the tracks on the CD?1. 160 nm 2. 1600 nm 3. 16µm 4. 160µmCheck YourselfWhat is the spacing of the tracks on the CD?grating tan θ θ sin θ D =500 nmsin θmanufacturing spec.CD130.32 0.31 1613 nm 1600 nmCheck YourselfDemonstration.laser pointerλ = 500 nmCDscreen3 feet1 feetWhat is the spacing of the tracks on the CD? 2.1. 160 nm 2. 1600 nm 3. 16µm 4. 160µmFourier Transforms in Physics: DiffractionDVD demonstration.Check
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