# Exercise 1.3: Rectangular Functions for Transmitter and Receiver

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Three different system configurations

We consider here three variants of a binary bipolar AWGN transmission system which differ with respect to the basic transmission pulse  $g_{s}(t)$  as well as the impulse response  $h_{\rm E}(t)$  of the receiver filter:

• For  $\text{System A}$,  both  $g_{s}(t)$  and  $h_{\rm E}(t)$  are rectangular,  only the pulse heights  $(s_{\rm 0}$  and  $1/T)$  are different.
• $\text{System B}$  differs from  $\text{System A}$  by having a triangular-shaped basic transmission pulse with  $g_{s}(t=0) = s_{\rm 0}$.
• $\text{System C}$  has the same rectangular basic transmission pulse as  $\text{System A}$,  while the impulse response is triangular with  $h_{\rm E}(t=0) = 1/T$.

The absolute width of the rectangular and triangular functions considered here is  $T = 10 \ \rm µ s$  each.  The bit rate is  $R = 100 \ \rm kbit/s$.  The other system parameters are given as follows:

$$s_0 = 6 \,\,\sqrt{W}\hspace{0.05cm},\hspace{0.3cm} N_{\rm 0} = 2 \cdot 10^{-5} \,\,{\rm W/Hz}\hspace{0.05cm}.$$

Notes:

$$\sigma _d ^2 = \frac{N_0 }{2} \cdot \int_{ - \infty }^{ + \infty } {\left| {H_{\rm E}( f )} \right|^2 \hspace{0.1cm}{\rm{d}}f} = \frac{N_0 }{2} \cdot \int_{ - \infty }^{ + \infty } {\left| {h_{\rm E}( t )} \right|^2 \hspace{0.1cm}{\rm{d}}t}\hspace{0.05cm}.$$

### Questions

1

Calculate for  $\text{System A}$  the basic detection pulse  $g_{d}(t) = g_{ s}(t) \star h_{\rm E}(t)$.  What value  $g_0 = g_{d}(t=0)$  results at time  $t = 0$?

 $g_0 \hspace{0.28cm} = \$ $\ \rm W^{1/2}$

2

From this,  calculate the detection noise power  (variance)  $σ_{d}^2$.

 $σ_{d}^{\hspace{0.02cm}2} \hspace{0.2cm} = \$ $\ \rm W$

3

Thus,  what is the bit error probability  $p_{\rm B}$  for  $\text{System A}$?

 $p_{\rm B} \hspace{0.2cm} = \$ $\ \cdot 10^{-9}$

4

Determine the corresponding quantities for  $\text{System B}$ .

 $g_0 \hspace{0.28cm} = \$ $\ \rm W^{1/2}$ $σ_{d}^{\hspace{0.02cm}2} \hspace{0.2cm} = \$ $\ \rm W$ $p_{\rm B} \hspace{0.2cm} = \$ $\ \cdot 10^{-2}$

5

What are the characteristics for  $\text{System C}$ ?

 $g_0 \hspace{0.28cm} = \$ $\ \rm W^{1/2}$ $σ_{d}^{\hspace{0.02cm}2} \hspace{0.2cm} = \$ $\ \rm W$ $p_{\rm B} \hspace{0.2cm} = \$ $\ \cdot 10^{-7}$

### Solution

#### Solution

1.  For System A,  the convolution of the two equal-width rectangular functions  $g_{s}(t)$  and  $h_{\rm E}(t)$  leads to a triangular detection pulse with the maximum at  $t = 0$:

$$g_d (t = 0) = \int_{ - T/2}^{ + T/2} { g_s(t) \cdot h_{\rm E}( t )} \hspace{0.1cm}{\rm{d}}t =s_0 \cdot \frac{1 }{T} \cdot T = s_0 \hspace{0.1cm}\underline { = 6 \,\,\sqrt{{\rm W}}}\hspace{0.05cm}.$$

There is no intersymbol interfering because for  $| t |\ge T$  the detection pulse is  $g_{d}(t) = 0$.

2.  The variance of the noise component of the detection signal – referred to as the  "detection noise power" – can be calculated in both the time and frequency domains.

• For the present rectangular waveform,  calculation in the time domain yields faster results:
$$\sigma _d ^2 \ = \ \frac{N_0 }{2} \cdot \int_{ - \infty }^{ + \infty } {\left| {h_{\rm E}( t )} \right|^2 \hspace{0.1cm}{\rm{d}}t} =\frac{N_0 }{2} \cdot \int_{ - T/2 }^{ + T/2 } {\left| {h_{\rm E}( t )} \right|^2 \hspace{0.1cm}{\rm{d}}t} = \ \frac{N_0 }{2} \cdot\frac{1 }{T^2} \cdot T = \frac{N_0 }{2T} = \frac{2 \cdot 10^{-5} \,\,{\rm W/Hz}}{2 \cdot 10^{-5} \,\,{\rm s}} \hspace{0.1cm}\underline {= 1\,{\rm W}}\hspace{0.05cm}.$$
• The frequency domain calculation would be as follows with  $H_{\rm E}(f) = {\rm sinc}(fT)$:
$$\sigma _d ^2 = \frac{N_0 }{2} \cdot \int_{ - \infty }^{ + \infty } {\left| {H_{\rm E}( f )} \right|^2 \hspace{0.1cm}{\rm{d}}f} = \frac{N_0 }{2} \cdot \int_{- \infty }^{ \infty } {\rm sinc}^2(f T)\hspace{0.1cm}{\rm{d}}f = \frac{N_0 }{2T} \hspace{0.05cm}.$$

3.  Due to the time-limited pulse shape  (this means:  no intersymbol interfering!),  the bipolar approach assumed here yields:

$$p_{\rm B} = {\rm Q} \left( \frac{s_0}{\sigma_d}\right)= {\rm Q} \left( \frac{ 6 \,\sqrt{\rm W}}{1 \,\sqrt{\rm W}}\right) = {\rm Q}(6) \hspace{0.1cm}\underline {= 0.987 \cdot 10^{-9}} \hspace{0.05cm}.$$
• System A  represents the matched filter realization of the optimal binary receiver,  so the following equations would also be applicable:
$$E_{\rm B} = s_0^2 \cdot T = 36\, {\rm W} \cdot 10^{-5} {\rm s}\hspace{0.3cm} \Rightarrow \hspace{0.3cm} p_{\rm B} = {\rm Q} \left( \sqrt{\frac{2 \cdot E_{\rm B}}{N_0}}\right) ={\rm Q} \left( \sqrt{\frac{2 \cdot 36 \cdot 10^{-5}\,\, {\rm Ws}}{2 \cdot 10^{-5} \,\, {\rm Ws}}}\right)={\rm Q}(6) \hspace{0.05cm}.$$

4.  Since  System B  uses the same receiver filter as  System A,  the same detection noise power  $σ_{d}^2 = 1 \ \rm W$  is also obtained.

• However,  the basic detection pulse is now no longer triangular,  but has a more pointed shape.  At time $t = 0$ applies:
$$g_d (t = 0) = \frac{1}{T} \cdot \int_{ - T/2}^{ + T/2} { g_s(t) } \hspace{0.1cm}{\rm{d}}t = \frac{1}{T} \cdot \frac{s_0 }{2} \cdot T = \frac{s_0 }{2}\hspace{0.1cm}\underline {= 3 \,\,\sqrt{\rm W}}\hspace{0.05cm}.$$
• System B  is also free of intersymbol interfering.  Therefore,  one obtains for the bit error probability:
$$p_{\rm B} = {\rm Q} \left( \frac{g_d (t = 0)}{\sigma_d}\right)= {\rm Q} \left( \frac{ 3 \,\sqrt{\rm W}}{1 \,\sqrt{\rm W}}\right) = {\rm Q}(3) \hspace{0.1cm}\underline {= 0.135 \cdot 10^{-2}} \hspace{0.05cm}.$$
• On the other hand,  the following calculation is  not  applicable here:
$$E_{\rm B} = \int^{+\infty} _{-\infty} g_s^2(t)\,{\rm d}t = 2\cdot s_0^2 \cdot \int ^{+T/2} _{0} \left( 1- \frac{2t}{T}\right)^2\,{\rm d}t = \frac{s_0^2 \cdot T }{3} = 12 \cdot 10^{-5} \,{\rm Ws}$$
$$\Rightarrow \hspace{0.3cm} p_{\rm B} = {\rm Q} \left( \sqrt{\frac{2 \cdot E_{\rm B}}{N_0}}\right) ={\rm Q} \left( \sqrt{12}\right)={\rm Q}(3.464) \approx 3 \cdot 10^{-4} \hspace{0.05cm}.$$
• One would thus compute a bit error probability that is too low,  since the implicit assumption of a matched filter does not hold.

5.  For the rectangular basic transmission pulse and the triangular impulse response   ⇒   System C,
the same basic detection pulse is obtained as for the triangular $g_{\rm s}(t)$ and the rectangular $h_{\rm E}(t)$.

• Therefore,  as in  System B:
$$g_d (t = 0) = \frac{s_0}{2}\hspace{0.1cm}\underline {= 3 \,\,\sqrt{\rm W}}\hspace{0.05cm}.$$
• In contrast,  the detection noise power is now smaller than in systems  A  and  B:
$$\sigma _d ^2 = \frac{N_0}{2} \cdot \frac{1}{T^2} \cdot \int^{+T/2} _{-T/2} \left( 1- \frac{2t}{T}\right)^2\,{\rm d}t = \frac{N_0}{6T}\hspace{0.1cm}\underline { = 0.333 \,{\rm W}}.$$
• This now gives us for the bit error probability:
$$p_{\rm B} = {\rm Q} \left( \frac{ 3 \,\sqrt{\rm W}}{0.577 \,\sqrt{\rm W}}\right) \approx {\rm Q}(5.2)\hspace{0.1cm}\underline { \approx 10^{-7} } \hspace{0.05cm}.$$
• The apparent increase in error probability by a factor of about  $100$  compared to subtask  (3)  is due to the severe mismatch compared to the matched filter.
• The improvement over subtask  (4)  is due to the higher signal energy.