6 003 Signals and Systems CT Fourier Transform April 8 2010 CT Fourier Transform Representing signals by their frequency content Z x t e j t dt analysis equation X j Z 1 x t X j ej t d 2 synthesis equation generalizes Fourier series to represent aperiodic signals equals Laplace transform X s s if ROC includes j axis inherits properties of Laplace transform complex valued function of real domain simple inverse relation more general than table lookup method for inverse Laplace duality filtering applications in physics Filtering Notion 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 R vi C vo Lowpass Filter Calculate the frequency response of an RC circuit R vi t Ri t vo t C i t C v o t Solving vi t vo C RC v o t vo t Vi s 1 sRC Vo s 1 Vo s H s Vi s 1 sRC 1 H j 0 1 0 01 H j vi KVL 0 01 0 1 1 10 100 1 RC 10 100 1 RC 0 2 0 01 0 1 1 Lowpass Filtering Let the input be a square wave 1 2 0 21 1 j 0 kt e j k k odd 1 X j X 0 2 T 0 1 0 01 X j x t t T 0 01 0 1 1 10 1 RC 100 10 100 1 RC 0 2 0 01 0 1 1 Lowpass Filtering Low frequency square wave 0 1 RC 1 2 0 21 1 j 0 kt e j k k odd 1 H j X 0 2 T 0 1 0 01 H j x t t T 0 01 0 1 1 10 1 RC 100 10 100 1 RC 0 2 0 01 0 1 1 Lowpass Filtering Higher frequency square wave 0 1 RC 1 2 0 21 1 j 0 kt e j k k odd 1 H j X 0 2 T 0 1 0 01 H j x t t T 0 01 0 1 1 10 1 RC 100 10 100 1 RC 0 2 0 01 0 1 1 Lowpass Filtering Still higher frequency square wave 0 1 RC 1 2 0 21 1 j 0 kt e j k k odd 1 H j X 0 2 T 0 1 0 01 H j x t t T 0 01 0 1 1 10 1 RC 100 10 100 1 RC 0 2 0 01 0 1 1 Lowpass Filtering High frequency square wave 0 1 RC 1 2 0 21 1 j 0 kt e j k k odd 1 H j X 0 2 T 0 1 0 01 H j x t t T 0 01 0 1 1 10 1 RC 100 10 100 1 RC 0 2 0 01 0 1 1 Source Filter Model of Speech Production Vibrations of the vocal cords are filtered by the mouth and nasal cavities to generate speech buzz from vocal cords throat and nasal cavities speech Filtering LTI systems filter signals based on their frequency content Fourier transforms represent signals as sums of complex exponentials Z 1 x t X j e j t d 2 Complex exponentials are eigenfunctions of LTI systems e j t H j e j t LTI systems filter signals by adjusting the amplitudes and phases of each frequency component Z Z 1 1 j t X j e d y t H j X j e j t d x t 2 2 Filtering Systems can be designed to selectively pass certain frequency bands Examples low pass filter LPF and high pass filter HPF LPF 0 HPF LPF t HPF t t Filtering Example Electrocardiogram An electrocardiogram is a record of electrical potentials that are generated by the heart and measured on the surface of the chest x t mV 2 1 0 1 t s 0 ECG and analysis by T F Weiss 10 20 30 40 50 60 Filtering Example Electrocardiogram In addition to picking up electrical responses of the heart electrodes on the skin also pick up a variety of other electrical signals that we regard as noise We wish to design a filter to eliminate the noise x t x t mV y t mV 2 1 0 1 t s 0 10 y t filter 20 30 40 50 60 2 1 0 1 t s 0 10 20 30 40 50 60 Filtering Example Electrocardiogram We can identify the noise by breaking the electrocardiogram into frequency components using the Fourier transform 1000 60 Hz X j V 100 10 1 0 1 0 01 low freq noise 0 001 cardiac signal 0 0001 0 01 0 1 1 10 f Hz 2 high freq noise 100 Filtering Example Electrocardiogram Filter design low pass flter high pass filter notch H j 1 0 1 0 01 0 001 0 01 0 1 1 10 Hz f 2 100 Electrocardiogram Check Yourself Which poles and zeros are associated with the high pass filter the low pass filter the notch filter s plane 2 2 2 Electrocardiogram Check Yourself Which poles and zeros are associated with the high pass filter the low pass filter the notch filter s plane notch low pass 2 notch 2 2 high pass Filtering Example Electrocardiogram By placing the poles of the notch filter very close to the zeros the width of the notch can be made quite small H j 1 0 5 0 59 60 f Hz 2 61 Filtering Example Electrocardiogram Comparision of filtered and unfiltered electrocardiograms y t mV x t mV 2 1 0 1 2 1 0 1 t s 0 10 20 30 40 50 1000 10 20 30 40 50 60 1000 60 Hz 100 Y j V 100 X j V t s 0 60 10 1 0 1 0 01 low freq noise 0 001 0 1 1 0 1 0 01 cardiac signal 0 0001 0 01 10 1 10 f Hz 2 high freq noise 100 0 001 0 0001 0 01 0 1 f 1 10 Hz 2 100 Filtering Example Electrocardiogram Reducing the frequency components that are not generated by the heart simplifies the output making it easier to diagnose cardiac problems x t mV Unfiltered ECG 2 1 t s 0 0 10 20 30 40 50 60 y t mV Filtered ECG 1 t s 0 0 10 20 30 40 50 60 Continuous Time Fourier Transform Summary Fourier 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 Transform Fourier transforms provide alternate views of signals 1 2 t 1 Pulses contain all frequencies except harmonics of 2 width 4 1 t 2 Wider pulses contain more low frequencies than narrow pulses 1 t Constants in time contain only frequencies at 0 2 Fourier Transforms in Physics Diffraction A diffraction grating breaks a laser beam input into …
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