Question

Calculate the inverse discrete time Fourier transform of:

H(jΩ) = { 1, |Ω| > Ωc 0, elsewhere

Ω is defined on the interval [−π,π]. (The resulting h[k] is the impulse response of an ideal highpass filter.)

Answer 1

The inverse discrete time Fourier transform of H(jΩ) = { 1, |Ω| > Ωc 0 is h[k] = (1/2πjk) (cos(πk) + jsin(πk) - e⁽ʲΩᵏ⁾)

Step-by-step explanation:

To calculate the inverse discrete-time Fourier transform (I D T F `T) of H(jΩ), we need to determine the impulse response h[k] of the ideal high pass filter.

The given transfer function H(jΩ) is defined as follows:

H(jΩ) = { 1, |Ω| > Ωc 0, elsewhere

To find the impulse response h[k], we will use the inverse Fourier transform:

1. Start with the definition of the inverse Fourier transform:

h[k] = (1/2π) ∫[−π,π] H(jΩ) e⁽ʲΩᵏ⁾ dΩ

2. Split the integral into two parts based on the given condition for H(jΩ):

h[k] = (1/2π) ∫[−π,−Ωc] 0 e⁽ʲΩᵏ⁾) dΩ + (1/2π) ∫[Ωc,π] 1 e⁽ʲΩᵏ⁾ dΩ

3. Evaluate the integrals:

h[k] = (1/2π) ∫[−π,−Ωc] 0 dΩ + (1/2π) ∫[Ωc,π] e⁽ʲΩᵏ⁾) dΩ

The first integral is zero, so we only need to calculate the second integral.

4. Solve the second integral:

h[k] = (1/2π) ∫[Ωc,π] e⁽ʲΩᵏ⁾) dΩ

Let's substitute u = jΩk, then dΩ = du/(jk).

h[k] = (1/2π) ∫[Ωc,π] eᵘ du/(jk)

5. Simplify the expression:

• h[k] = (1/2πjk) eᵘ ∣ [Ωc,π]
• h[k] = (1/2πjk) (e⁽ʲΩᵏ⁾ ∣ [Ωc,π])
• h[k] = (1/2πjk) (e⁽ʲπᵏ⁾ - e⁽ʲΩᵏ⁾)

6. Finally, simplify further:

h[k] = (1/2πjk) (cos(πk) + jsin(πk) - e⁽ʲΩᵏ⁾)

The above equation represents the impulse response h[k] of the ideal highpass filter.