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The Discrete-Time Fourier Transform

The Discrete-Time Fourier Transform. Objectives. To generalize the discrete-time Fourier to include aperiodic signals by defining the discrete-time Fourier transform. To establish which types of signals can or cannot be described by a discrete-time Fourier transform.

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The Discrete-Time Fourier Transform

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  1. The Discrete-Time Fourier Transform

  2. Objectives • To generalize the discrete-time Fourier to include aperiodic signals by defining the discrete-time Fourier transform. • To establish which types of signals can or cannot be described by a discrete-time Fourier transform. • To demonstrate the properties of the discrete-time Fourier transform. • To demonstrate the interrelationships among the Fourier methods.

  3. DTFS-to-DTFT Transition Pulse Train This periodic rectangular-wave signal is analogous to the continuous-time periodic rectangular-wave signal used to illustrate the transition from the CTFS to the CTFT.

  4. DTFS-to-DTFT Transition (cont…) • The DTFS harmonic function for this signal x[n] over all discrete time with NF = N0 is X[k] = ((2Nw + 1)/ N0)drcl(k/N0, 2Nw + 1)

  5. DTFS-to-DTFT Transition (cont…) DTFS of Pulse Train As the period, N0 (N0=NF) of the rectangular wave increases, the period of the DTFS increases but the amplitude of the DTFS decreases.

  6. DTFS-to-DTFT Transition (cont…) Normalized DTFS of Pulse Train

  7. DTFS-to-DTFT Transition (cont…) As N0 approaches infinity, the separation between points of N0 X[k] approaches zero and the discrete frequency graph becomes a continuous frequency graph.

  8. Definition of the DTFT F Form Inverse Forward Inverse Forward W Form

  9. DTFT Properties Linearity

  10. DTFT Properties (cont…) Time Shifting Frequency Shifting Time Reversal

  11. DTFT Properties (cont…) Time and Frequency Scaling There is no scaling property for this kind of time scaling because there is unique relationship between a signal and a decimated version of that signal. Two different signals can be decimated to form the same signal.

  12. DTFT Properties (cont…) Time and Frequency Scaling There is one kind of time scaling for which there is a time scaling property. This also implies a limited form of frequency scaling. If F is changed to mF (m an integer), the effect in the time domain is the insertion of m-1 zeros between adjacent points as described above.

  13. DTFT Properties (cont…) Differencing Accumulation

  14. DTFT Properties (cont…) Multiplication- Convolution Duality As is true for other transforms, convolution in the time domain is equivalent to multiplication in the frequency domain

  15. DTFT Properties (cont…) Accumulation Definition of a Periodic Impulse Parseval’s Theorem The signal energy is proportional to the integral of the squared magnitude of the DTFT of the signal over one period.

  16. The Four Fourier Methods

  17. Relations Among Fourier Methods

  18. Relations Among Fourier Methods Multiplication-Convolution Duality

  19. Relations Among Fourier Methods Parseval’s Theorem

  20. Relations Among Fourier Methods Time and Frequency Shifting

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