Time and Frequency Characterization of Signals & Systems
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Transcript of Time and Frequency Characterization of Signals & Systems
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Time and Frequency Characterization of Signals & Systems
•Frequency Domain Characterization through multiplication of FourierTransform of input signal and system frequency response.( Transfer Function).
•Time Domain Characterization through convolution of input signal and system impulse response.
Convenient to use frequency domain because easy operation of Multiplication as oppose to operation of convolution in time domain.
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Magnitude and Phase Representation of Fourier Transforms
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j
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)(X je)(e H )X(e)Y(e jjj
)(X)(H)(Y jjj
Magnitude-Phase Representation of The Frequency Response of LTI Systems
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x(t) y(t)=h(t)*x(t)
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Linear Phase and Group Delay of LTI Systems
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Log-Magnitude and Bode plots
The absolute values of the magnitude of the transfer function of a system are normally converted into decibels define as
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Continuous-time Filters Described By Differential Equations.
Simple RC Lowpass Filter.
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+ -
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Continuous-time Filters Described By Differential Equations.
Simple RC Lowpass Filter.
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+ -
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First-order Recursive Discrete-time Filter
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First-order Recursive Discrete-time Filter
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1)H(e
)(
)Y(
)()(ae-)Y(
-:equation above of DTFT Taking
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XY
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D
+
a
x[n] y[n]
y[n-1]
ay[n-1]
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x[n] y[n]=x[n]*h[n]X( Y(X(
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First-order Recursive Discrete-time Filter
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)H(e][
anua
nhTF
n
TF
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Impulse response of First order recursive D-T lowpass filter
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0
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Frequency Response of First order recursive lowpass filter
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Impulse-Train Sampling
n
nTttp )()(
n
p nTtnTxtptxtx )()()()()(
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Multiplication/Modulation Property
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TF
p )(()(2
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00
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p
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Convolution in Frequency Domain
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Sampling Theorem
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Violating Sampling Theorem resulting in aliasing.
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Reconstruction using an ideallowpass filter.
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Reconstruction looking from the time domain-convolving.
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H(j Y(jjX(j
)(txp )(*)()( thtxty p
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).(}.)()({)(
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)()(
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Continuous to discrete-time signal conversion.
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Discrete-time Processingof Continuous-time Signals.
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Reconstruction of a sampled signal with a zero-order hold
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Comparison of Frequency Responses (Transfer Functions) of ideal lowpass reconstruction filter and zero-order hold
reconstruction filter.
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Reconstruction of a sampled signal with a first-order hold
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Comparison of Frequency Responses (Transfer Functions) of ideal lowpass reconstruction filter, zero-order hold
reconstruction filter and first-order hold reconstruction filter.
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Reconstruction of a sampled signal with ideal lowpass filter