Frequency Response with Delta Function?

Question

I am trying to find frequency response and magnitude of the frequency response of the following system impulse response: $$h[n] = 2\delta [n] + 2\delta [n-1]$$

I understand, that through the DTFT: $$H(\omega ) = 2(1 + e^{-j\omega})$$

However, I am unsure how to find the magnitude of the frequency response. The solution is: $$|H(\omega )| = 4 \left| \cos \left(\frac{\omega }{2} \right) \right|$$

What steps are needed in order to convert the above frequency response to the magnitude of the frequency response?

Answer 1

The long road computes the modulus of the DTFT. This works in general, but can be tedious.

When the formula possesses some symmetry, like here (the two coefficients are the same), you can more efficiently factor a term, so that you recover a known Euler formula for the sine or the cosine, in the shape of $e^{j\nu}+e^{-j\nu}$ or $e^{j\nu}-e^{-j\nu}$. Here, factoring by $e^{-j\omega/2}$ yields $2e^{-j\omega/2}(e^{j\omega/2}+e^{-j\omega/2})$, from which the result follows easily.

Answer 2

Hint: Expand $e^{j \omega}$ using Euler's formula. This will enable you to express $H(\omega) = R(\omega) + j I(\omega)$ where $R$ and $I$ are the real and imaginary parts and then the magnitude of the frequency response is just $\sqrt{ R(\omega)^2 + I(\omega)^2}$.