Question

Consider a periodic signal x(t) with period Tₚ=2s (i.e. x(t) = x(t+nTₚ) for any integer n), given by:

x(t) = -1, for -1(t) = t, for 0This signal is input to a 256-level uniform quantizer with a dynamic range of 1.5 volts.
(a) Sketch x(t) for -2(b) Find the SQNR (in dB) at the quantizer output when the input is x(t).

Answer 1

The problem requires sketching a piecewise periodic signal and calculating its Signal-to-Quantization-Noise Ratio (SQNR) when quantized by a 256-level uniform quantizer. The signal sketch would exhibit step and ramp segments, while the SQNR calculation for 8-bit quantization results in 49.92 dB.

Step-by-step explanation:

Periodic Signal Quantization and SQNR Calculation

The problem involves a periodic signal x(t) with a known period of 2 seconds and a piecewise definition. To sketch x(t) for one period, we draw a horizontal line at x = -1 from t = -1 to t = 0, and then a line with unit slope from t = 0 to t = 1. No sketch can be provided here because this is a text-based answer.

To calculate the Signal-to-Quantization-Noise Ratio (SQNR), we take into account that the signal is quantized to 256 levels, meaning there are 255 quantization steps in the dynamic range of 1.5 volts. Using quantization theory, the SQNR in decibels (dB) can be calculated using SQNR = 1.76 + 6.02 × number of bits. Since the number of levels is 256, we have 8 bits (since 28 = 256). Consequently, the SQNR is 1.76 + 6.02 × 8 = 49.92 dB.

Note that the determination of SQNR is based on the assumption of the signal being full scale within the quantizer's dynamic range and that the quantization noise is uniform and uncorrelated with the signal.