(CPE 531 ) DSP Mock Exam

Consider a discrete-time system described by the input-output relationship: $y[n] = n \cdot x[n] + x[n-1]$ where $x[n]$ is the input and $y[n]$ is the output.

a) Determine if the system is linear. Provide a step-by-step demonstration of your conclusion.

b) Investigate if the system is time-invariant. Clearly show your working.

c) Determine whether the system is causal. Explain your reasoning.

d) For an LTI system, stability is determined by its impulse response. Consider an LTI system with the impulse response $h[n] = (0.5)^n u[n]$. i) Determine if this system is stable. Justify your answer using the appropriate stability criterion. ii) If the input to the system from part (d) is $x[n] = \delta[n-1]$, calculate the output $y[n]$.

A continuous-amplitude analog signal, $x(t)$, is sampled and then quantized using a 3-bit quantizer. The sampled values of $x[n]$ range over the interval $[-4V, 4V]$.

a) Calculate the step size ($\Delta$) for this quantizer. Determine all the quantization levels ($X_q$) and their corresponding binary codes. Present your answer in a tabular format showing the level index ($q$), decision interval, quantization level ($X_q$), and binary code.

b) Given the sequence of sampled signals $x[n] = [-3.2V, 0.8V, 2.1V, -1.9V, 3.7V]$, quantize these values using the levels determined in part (a). For each sample, state its quantized value and calculate the quantization error ($e_q = X_q - X$).

c) If the number of bits used for quantization were increased from 3 bits to 4 bits, explain how this change would affect the quantization noise power.

A continuous-time analog signal, $x(t)$, is to be converted into a digital signal using an Analog-to-Digital Converter (ADC). The signal's amplitude is defined over a range of $-10V$ to $10V$.

a) If a 4-bit ADC is used for quantization, determine the following: i) The total number of quantization levels, $L$. ii) The step size (quantum), $\Delta$, for uniform quantization. iii) The quantization levels, $X_q$, for each index $q$.

b) Suppose a sequence of sampled values from the analog signal is $x[n] = [ -7.2V, 0.8V, 4.5V, -3.1V, 9.6V ]$. i) Quantize each sample value to its corresponding representation level determined in part (a). ii) Calculate the quantization error, $e_q = X_q - X$, for each quantized sample.