How do I deal with a seemingly fractional delays in discrete time fourier transforms?

Question

Is a transfer function of a discrete time system is $H(e^{j\Omega})=e^{-j\Omega/4}$ and I feed it an impulse, what will be it's response? I know that technically a transfer function of $e^{j{\Omega}n_0}$ represents a delay of $n_0$ samples. What happens if the delay is fractional?

Answer 1

Simply compute the inverse discrete-time Fourier transform:

$$h(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi}H(e^{j\Omega})e^{jn\Omega}d\Omega$$

With $H(e^{j\Omega})=e^{-jr\Omega}$ ($r$ not integer) you get

$$h(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{j\Omega (n-r)}d\Omega=\frac{1}{2\pi j(n-r)}\left(e^{j\pi (n-r)}-e^{-j\pi (n-r)}\right)=\frac{\sin\pi(n-r)}{\pi (n-r)}$$

So you get a sampled sinc function shifted by $r$.