Sampling theory revisited sampling theory revisited

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Sampling Theory Revisited Sampling Theory Revisited USRP Data Flow Representation of Sampling I Mathematically convenient to represent in two stages I Impulse train modulator I Conversion of impulse train to a sequence Convert impulse train to discrete- time sequence Xc(t) s(t) X[n]=Xc(nT) Figure: Conversion of Samples to Analog Waveforms. Professor Alexander M. Wyglinski ECE4305: Software-Defined Radio Systems and Analysis
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Sampling Theory Revisited Sampling Theory Revisited USRP Data Flow Continuous-Time Fourier Transform I Continuous-Time Fourier transform pair is defined as I X c ( j Ω) = R -∞ x c ( t ) e - j Ω t dt I x c ( t ) = 1 2 π R -∞ X c ( j Ω) e j Ω t d Ω I We write x c ( t ) as a weighted sum of complex exponentials I Remember some Fourier Transform properties I x ( t ) * y ( t ) X ( j Ω) Y ( j Ω) I x ( t ) y ( t ) X ( j Ω) * Y ( j Ω) I x ( t ) e j Ω 0 t X ( j - Ω 0 )) Professor Alexander M. Wyglinski ECE4305: Software-Defined Radio Systems and Analysis
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Sampling Theory Revisited Sampling Theory Revisited USRP Data Flow Frequency Domain Representation of Sampling I Modulate (multiply) continuous-time signal with pulse train: I x s ( t ) = x c ( t ) s ( t ) = n = -∞ x c ( t ) δ ( t - nT ) s ( t ) = n = -∞ δ ( t - nT ) I Let’s take the Fourier Transform of x s ( t ) and s ( t ) I X s ( j Ω) = 1 2 π X c ( j Ω) * S ( j Ω) S ( j Ω) = 2 π T k = -∞ δ - k Ω s ) I Fourier transform of pulse train is again a pulse train I Note that multiplication in time is convolution in frequency I We represent frequency with Ω = 2 π f , hence Ω s = 2 π f s I X s ( j Ω) = 1 T k = -∞ X c ( j - k Ω s )) Professor Alexander M. Wyglinski ECE4305: Software-Defined Radio Systems and Analysis
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Sampling Theory Revisited Sampling Theory Revisited USRP Data Flow Frequency Domain Representation of Sampling I Convolution with pulse creates replicas at pulse location: I X s ( j Ω) = 1 T k = -∞ X c ( j - k Ω s )) I This tells us that the impulse train modulator I Creates images of the Fourier transform of the input signal I Images are periodic with sampling frequency I If Ω s < Ω N , sampling maybe irreversible due to aliasing of images Professor Alexander M. Wyglinski ECE4305: Software-Defined Radio Systems and Analysis
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