Question

The z-transform of a discrete-time signal is defined H(z)=∑h[n]z⁻ⁿ If signal h[n] is the impulse response of a filter, the frequency response of that filter is determined by evaluating H(z) along the unit circle according to

H(eʲΩ)=∑h[n]e⁻ʲΩⁿ
with inverse transform h[n]=
1/2π∫₂πH(eʲΩ)eʲΩⁿdΩ
The frequency response can be determined at frequencies Ωₖ = 2πk/Naccording to H(eʲΩᵏ)=Hₙ[k] where DFT
Hₙ[k]=0
∑ Hₙ[n]e⁻ʲ²πⁿᵏ/ᴺ
N−1
​
for periodic extension
Hₙ[n]=∑ h[n−mN]
m
​
Determine a formula for the ideal low-pass filter hᵢ[n] where Hᵢ(eʲΩ)=1 for ∣Ω−2πm∣<2πfT

Answer 1

To determine the formula for the ideal low-pass filter hᵢ[n], we can use the concept of the discrete Fourier transform (DFT). By calculating the inverse DFT of Hᵢ(eʲΩ), we can obtain the filter's impulse response. Evaluating H(eʲΩₖ) at the given frequencies Ωₖ = 2πk/N allows us to determine Hₙ[k] for the periodic extension Hₙ[n].

Step-by-step explanation:

To determine a formula for the ideal low-pass filter hᵢ[n], where Hᵢ(eʲΩ)=1 for ∣Ω−2πm∣<2πfT, we can use the concept of the discrete Fourier transform (DFT).

We can calculate the inverse DFT of Hᵢ(eʲΩ) to obtain the filter's impulse response, which is the desired formula for hᵢ[n].

By plugging in the given values for frequencies Ωₖ = 2πk/N, where N is the size of the DFT, we can evaluate H(eʲΩₖ) to get Hₙ[k].

For the periodic extension Hₙ[n], we can use the formula Hₙ[n]=∑ h[n−mN] m, where m ranges from negative infinity to infinity.