Relationship between DFT and Z transform

We already learn about what is DFT and what is Z transform, So now here this article gives the information about the relationship between DFT and Z transform to know more details about DFT as well as Z transform.

The Z transform of sequence x(n) is,

X(Z) = ∑  x(n) . Z –n

We know that at Z = e –jω

X(Z) = ∑  x(n) . e jωn
           n = -∞

It means that X(Z) is evaluated on the unit circle. 

Now suppose X(Z) is sampled at N equally spaced point on the unit circle, then we have 

ω = 2πK / N

Now if X(Z) is evaluated at Z =  e jωk/n then by putting equation we get; 

X(Z) = ∑  x(n) . e-j2πKn / N
           n = -∞

At Z = ej2πK/N

In the equation, if x(n) is a causal sequence and has N number of the sample then we can write an equation

X(K) = ∑   x(n) . e-j2πKn / N
           n = 0

At Z = ej2πKn/N

X(K) = X(Z)  At Z = ej2πKn/N