• Domain of DFT can be integers.

  • Discrete Fourier Series (DFS) = Discrete Fourier Transform (DFT)

  • The Discrete Fourier Series (DFS) is the version of the Fourier series used when your signal is periodic and discrete-time (sampled). It expresses a discrete-time periodic signal as a sum of complex exponentials with integer frequency multiples.

  • the DFS maps an N-periodic signal onto an N-periodic sequence of Fourier coefficients.

  • the inverse DFS maps an N-periodic sequence of Fourier coefficients a set onto an N-periodic signal

  • the DFS of an N-periodic signal is mathematically equivalent to DFT of one period

  • M-length signal y[n] with L repetition.

• Discrete time Fourier transform (DTFT)

  • From the DFT, as N goes to infinity, the frequency (2pi/N) goes to smaller and smaller.

  • Let's substitute (2pi/N)k = omega
    
    
    
    
    
    
    

  • F(w) is 2pi - periodic

  • To emphasize periodicity, we will write

Examples

• Existence

• DTFT Properties

  • Linearity:

  • Time shift:

  • Modulation (dual):

  • Time reversal:

  • Conjugation

• Some special cases

  • if x[n] is symmetric, the DTFT is symmetric

  • if x[n] is real, the DTFT is Hermitian symmetric

  • if x[n] is real, the magnitude of the DTFT is symmetric.

  • if x[n] is real and symmetric, X(e^{j\omega}) is also real and symmetric.

• Is Everything summable?

So this will introduce a dirac-delta function.

The direc delta function is summable, So we can use the direc delta function to compute the DTFT.

• Sinusoidal modulation

• Embedding finite-length signals

  • periodic extension

  • finite-support extension.