Signal to Quantization Noise Ratio (SQNR)

Question

I have this problem in my homework (following picture).

How do find the signal to quantization noise ratio at the output?

I don't want the solution. I just want to know a better approach to tackle it.

Because the way I am trying to do is - take the inverse fourier transform of H(z) and convolve it with x(n) and find the power. Then divide this recently found power by the noise power. As easy it sounds (probably correct) it is computationally intense.

I end up with 4 convolutions just to find the y(n).

Answer 1

No need to do the convolution to find the output. Just use the fact that $z^n$ is an eigenfunction of the system with $z=e^{j\omega_0}$. This will lead to the following rule

$$x[n]=A\sin(\omega_0 n+\phi) \Rightarrow y[n]=|H(e^{j\omega_0})|A \sin(\omega_0 n+\phi+\angle H(e^{j\omega_0}))$$

That is, the output to a sinusoidal signal with angular frequency $\omega_0$ is a similar sinusoidal signal with the same angular frequency, but scaled by $|H(e^{j\omega_0})|$ and phase-shifted by $\angle H(e^{j\omega_0})$.

For SQNR calculation, we only need the scaling $|H(e^{j\omega_0})|$. You probably know how to find the power of a sinusoidal signal, given its amplitude. Right?

Then you need to know how to calculate SQNR when a sinusoidal signal with amplitude $|H(e^{j\omega_0})|A$ gets quantized.