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
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
N-1
X(K) = ∑ x(n) . e-j2πKn / N
n = 0
At Z = ej2πKn/N
X(K) = X(Z) At Z = ej2πKn/N
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
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
N-1
X(K) = ∑ x(n) . e-j2πKn / N
n = 0
At Z = ej2πKn/N
X(K) = X(Z) At Z = ej2πKn/N