Discrete-Time Fourier Transform
• Continuous-Time Fourier Transform
• Discrete-Time Fourier Transform.
• Discrete-Time Fourier Transform Theorems.
• Band-Limited Discrete-Time Signals
• The Frequency Response of an LTI Discrete-Time System
• Phase and Group Delays
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• CTFT of continuous-time signal will only exist if the
Dirichlet conditions are satisfied:-
• (1) Signal must have finite number of discontinuities
and finite number of maximum & minimum in any
finite interval of time.
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Properties of Fourier Transform
• Linearity
• Time Shifting
• Conjugation
• Differentiation in the time-domain
• Integration in the time-domain
• Time and Frequency Scaling.
• Duality
• Parseval’s Relation
• Convolution
• Multiplication
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Concept of Filtering
• Discrete Signals are sequences of numbers in time-domain.
• Through DTFT, Discrete Signals are transformed into frequency domian which are represented as magnitude and phase spectrum of frequencies.
• Shape the Spectrums to the desired forms by multiplying them with frequency response of LTI Discrete-Time Systems.– Frequency shaping filters
– Frequency selective filters.
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What happen to Phase?
• Phase and Group Delays need to be considered
since both DTFT of Input and impulse response
are complex functions.
• Multiplying these two complex functions will
result in addition of their phase components
(argument of complex function)
• Group delay is defined as the negative
derivative of phase function with respect to
angular frequency.
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