### 1. Relationship between continuous and discrete signals

## Task

A complex-valued continuous-time signal \(v_a(t)\) has the Fourier transform shown in the figure 1.

Figure 1: Fourier transform of \(v_a(t)\)

This signal is sampled to produce the sequence \(v(n)=v_a(nT)\).

- Sketch the Fourier transform \(V(e^{j\Omega})\) of the sequence \(v(n)\) for \(T=\frac{\pi}{\omega_2}\).
- What is the lowest sampling frequency that can be used without incurring any aliasing distortion, i.e. so that \(v_a(t)\) can be recovered from \(v(n)\)?

## Amount and difficulty

- Working time: approx. 20 minutes
- Difficulty: easy

### 2. Overall system for filtering a continuous-time signal in digital domain

## Task

Figure 2 shows an overall system for filtering a continuous-time signal using a discrete-time filter. The frequency response of the ideal reconstruction filter \(H_\textrm{r}(j\omega)\) and the discrete-time filter are shown below.

Figure 2: Overall system

- For \(V_\textrm{a}(j\omega)\) as shown in figure 3 and \(1/T=20kHz\) sketch \(V_\textrm{i}(j\omega)\) and \(V(e^{j\Omega})\).

Figure 3: Spectrum of \(V_\textrm{a}(j\omega)\) and \(H_\textrm{eff}(j\omega)\)

For a certain range of values of T, the overall system, with input \(v_\textrm{a}(t)\) and output \(y_\textrm{r}(t)\), is equivalent to a continuous-time lowpass filter with frequency response \(H_\textrm{eff}(j\omega)\) sketched in figure 3.

- Determine the range of values of \(T\) for which the information presented above is true, when \(V_\textrm{a}(j\omega)\) is

bandlimited to \(|\omega | \leq 2\pi \cdot 10^4\) as shown in figure 3. - For the range of values determined in (b), sketch \(\omega_c\) as a function of \(1/T\).

**Note:** This is one way of implementing a variable-cutoff continuous-time filter using fixed continuous-time and discrete-time filters and a variable sampling rate.

## Amount and difficulty

- Working time: approx. 30 minutes
- Difficulty: easy

### 3. Quantization

## Task

A sinusoid signal \(v(n) = 5 \sin (\frac{\omega_0}{\omega_{\textrm{s}}} \cdot n)\) with \(f_0=5\textrm{ Hz and } f_{\textrm{s}}=10\textrm{ kHz}\) has to be quantized \(\textrm{(}v_{\textrm{q}} = Q[v(n)]\textrm{)}\) with a midtreat quantizer. The range of the signal is \(\pm 5\) V and the word length of the quantizer 4 bits. The quantizer at digital full scale.- How many quantization levels \(L\) does the quantizer have? What is the value of \(\Delta\)?
- Sketch the input-output characteristic of the quantizer. How different is a midtreat quantizer to a midrise quantizer.
- For time index \(n = 1250\) calculate the quantized value \(v_{\textrm{q}}(n)\), the quantization error \(e_{\textrm{q}}(n)\) and represent \(v_{\textrm{q}}(n)\) using bipolar code (sign and magnitude representation).

The quantization error over time can be modeled as a noise that is added to the input signal.

- Sketch the real system and the mathematical model of the system with the added quantization noise.
- Calculate the power \(P_{\textrm{n}}\) of the quantization noise.

Determine the SNR in dB and in The signal's amplitude is changed to \(\pm 1\) V, while the range \(R\) of the quantizer remains

- How is SNR affected with this change?
- What word length has to be chosen to achieve an SNR > 45 dB?