Exploring Discrete Space Fourier Transform (DSFT)

Understanding DSFT of Discrete Delta Functions

The student's question is about finding the Discrete Space Fourier Transform (DSFT) of two discrete delta functions, δ(n₁,n₂) and δ(n₁−5,n₂−3). The DSFT of a two-dimensional discrete delta function δ(n₁,n₂) is a constant in the frequency domain because the delta function is nonzero only at the origin (n₁=0, n₂=0) and zero otherwise. This results in a frequency response that remains the same for all frequencies, which is the definition of a constant function in the frequency domain.

For the DSFT of δ(n₁−5,n₂−3), this would essentially be a shifted delta function in the spatial domain, which results in a complex exponential term in the frequency domain that corresponds to its position shift. As a general rule, a spatial shift in the time domain corresponds to a phase shift in the frequency domain, which can be captured by multiplying the frequency response of the original signal by a complex exponential factor. The specific form of this factor would depend on the shift amounts and is derived from the properties of the Fourier Transform.