# Quantization problem

Following is the problem:

A sinusoidal signal $$A_c\cos(2πf_ct + φ)$$, where φ is uniformly distributed over $$[−π, π]$$, is quantized by a 1-bit quantizer. Assume that the signal utilizes all the representation levels of the quantizer. What will be the signal-to-quantization noise ratio?

I have done some basic problems on quantization but I am not sure how to proceed when a probability distribution is given.

Any help is appreciated. Thanks!

without loss of generality, it doesn't matter what $\varphi$ is and it can be assumed $A_c \ge 0$.

since this is a 1-bit quantizer, the sole quantizing threshold is at 0. then the error will depend on the ratio of the output level of the quantizer and $A_c$. let the 1-bit quantizer be defined as:

$$\hat{x}(t) \triangleq \frac{\Delta}{2}\operatorname{sgn}\{ x(t) \}$$

where $x(t) = A_c \sin(2 \pi f_c t + \varphi)$ and the stepsize $\Delta > 0$.

and the quantization error is

$$\epsilon_x(t) \triangleq \hat{x}(t) - x(t)$$

if the output level of the 1-bit quantizer is $\pm 1$ (which means $\Delta=2$) and $A_c$ is a million, you can expect an awful lotta quantization error. but this 1-bit quantizer will behave exactly the same as if $A_c = 2$ in which the quantization error would be far less. somehow, a 1-bit quantizer has an inherent gain factor in it and that has to be modeled from some knowledge of the nature of the amplitude of $x(t)$.

• Thanks for your comment but I still do not see how to solve the problem at hand. How do I go about finding the noise here? Sorry if I miss something obvious here. Thanks again! – Pranav Arora Nov 26 '16 at 8:29
• well, when $x(t)>0$, then $\hat{x}(t)=+\frac{\Delta}{2}$ and when $x(t)<0$, then $\hat{x}(t)=-\frac{\Delta}{2}$. see if you can come up with an expression for $\epsilon_x(t)$. then square it and find the average of the square. – robert bristow-johnson Nov 26 '16 at 18:36
• To find the average, I need to integrate it over something but what is the integration variable here? How do I factor in the distribution of $\varphi$ here? Can you please help me set up the integral for this? – Pranav Arora Nov 26 '16 at 18:43
• turns out that it doesn't matter what $\varphi$ is. you can set it to any number you want. the averaging of $|\epsilon_x(t)|^2$ is over time and you can do that averaging over one period. – robert bristow-johnson Nov 26 '16 at 18:45
• Ok, let $\varphi$ be zero. Then the integral is: $\frac{1}{T}\int_0^{T} (\hat{x}(t)-A_c\sin(2\pi f_c t))^2dt$. Solving this is not too hard but I fail to see why the distribution doesn't matter here. :/ – Pranav Arora Nov 26 '16 at 18:49