Relating the Discrete-Time Fourier Transforms (DTFTs)

• Fact: The 1-D Discrete-Time Fourier Transform of $g$ and the 2-D Discrete-Time Fourier Transform of $x$ are related as follows:
  
  $$G(\omega)=\sum^{N^2-1}_{n=0}g(n)e^{-j{\omega}n}$$
  
  
  Let $n = N n_1 + n_2, 0 \le n_1 < N, 0 \le n_2 < N$,
  
  $$G(\omega) = \sum^{N-1}_{n_1=0}\sum^{N-1}_{n_2=0} g(N n_1 + n_2) e^{-j\omega(N n_1 + n_2)}$$
  
  
  
  
  $$G(\omega) = \sum^{N-1}_{n_1=0} \sum^{N-1}_{n_2=0} x(n_1, n_2) e^{-j\omega{Nn_1}}e^{-j\omega{n_2}} =X(N\omega, \omega)$$
  
  
  Fig. 5 shows one scan line passing through the origin. Due to the periodicity of the Fourier domain, the scan line replicates every $2 pi$ along the $\omega_2$ axis, and every $\frac{2 pi}{N}$ along the $\omega_1$ axis.

  Figure 5: Relationship between the 2-D DTFT and the 1-D DTFT. Replicas of the scan line due to the periodicity of the 2-D Fourier domain are not shown.

  

• $G(\omega)$ is a scanned version of $X(\omega_1, \omega_2)$.
  Lexicographic ordering in $n \stackrel{\em F}{\longleftrightarrow}$ Scanning in $\omega$
  Scanning in $t_1, t_2 \stackrel{\em F}{\longleftrightarrow }$ Lexicographic ordering of 2-D Fourier series coefficients