Probability of Error MCQ Quiz - Objective Question with Answer for Probability of Error - Download Free PDF

Given:

Probability of error when  0 is received = p

Probability of error when  1 is received = 1 - p

If the number of zeros exceeds the number, the decoder decodes it as ‘0” otherwise as “ 1”. 

For n = 3 bits

Hence the probability of error is,

P_e = p³ + p²(1 - p) + p2(1 - p) + p2(1 - p) 

P_e = p³ + 3p2(1 - p)

When 1 is transmitted received signal is:

R = 1 + N

Probability of error $=\frac{1}{2}\left(1\right)\left(\frac{1}{4}\right)$

[Area of shaded region] = 1/8

When -1 is transmitted, the received signal is:

Probability of error = Area of the shaded region = 1/8

Total probability of error is:

P[E] = P(x = -1)P_E1 + P(x = 1) P_E2

$=\frac{1}{2}\left(\frac{1}{8}\right)+\frac{1}{2}\left(\frac{1}{8}\right)$

$=\frac{1}{8}=0.125$

Based on the information given for the two symbols waveforms, we see that the basis function is $\sqrt{\frac{2}{\mathrm{T}}}\sin \left({\omega }_{\mathrm{c}}\mathrm{t}\right)$. Now the oscillator frequency is $\sqrt{\frac{2}{\mathrm{T}}}\sin \left({\omega }_{\mathrm{c}}\mathrm{t}+{45}^{\circ }\right)$

Using this, we calculate the vectors of demodulated symbols of  ${\mathrm{s}}_1\left(\mathrm{t}\right)$ and ${\mathrm{s}}_2\left(\mathrm{t}\right)$ along the basis function and get the following result.

${\mathrm{S}}_1=\sqrt{\frac{\mathrm{E}}{2}}$and ${\mathrm{S}}_2=-\sqrt{\frac{\mathrm{E}}{2}}$

Now, distance between vectors $\mathrm{d}=2\sqrt{\frac{\mathrm{E}}{2}}$

Now, using the relation that ${\mathrm{P}}_{\mathrm{e}}=\mathrm{Q}\left(\sqrt{\frac{{\mathrm{d}}^2}{2{\mathrm{N}}_{\mathrm{o}}}}\right)=\mathrm{Q}\left(\sqrt{\frac{4\times \frac{\mathrm{E}}{2}}{2{\mathrm{N}}_{\mathrm{o}}}}\right)=\mathrm{Q}\left(\sqrt{\frac{\mathrm{E}}{{\mathrm{N}}_{\mathrm{o}}}}\right)$

Alternate Solution:

For BPSK

For coherent detection when there is phase synchronization between the carrier and received signal:

$P_e=Q\left(\sqrt{\frac{2E_b}{N_o}}\right)$

For coherent detection when there is phase shift of ϕ  between the carrier and received signal:

$P_e=Q\left(\sqrt{\frac{2E_{b}Co{s}^2\varphi }{N_o}}\right)$

Substituting the values Cos² ϕ = 1/2

$P_e=Q\left(\sqrt{\frac{E}{N_o}}\right)$

When 1 is transmitted received signal is:

R = 1 + N

Probability of error $=\frac{1}{2}\left(1\right)\left(\frac{1}{4}\right)$

[Area of shaded region] = 1/8

When -1 is transmitted, the received signal is:

Probability of error = Area of the shaded region = 1/8

Total probability of error is:

P[E] = P(x = -1)P_E1 + P(x = 1) P_E2

$=\frac{1}{2}\left(\frac{1}{8}\right)+\frac{1}{2}\left(\frac{1}{8}\right)$

$=\frac{1}{8}=0.125$

Given:

Probability of error when  0 is received = p

Probability of error when  1 is received = 1 - p

If the number of zeros exceeds the number, the decoder decodes it as ‘0” otherwise as “ 1”. 

For n = 3 bits

Hence the probability of error is,

P_e = p³ + p²(1 - p) + p2(1 - p) + p2(1 - p) 

P_e = p³ + 3p2(1 - p)

Based on the information given for the two symbols waveforms, we see that the basis function is $\sqrt{\frac{2}{\mathrm{T}}}\sin \left({\omega }_{\mathrm{c}}\mathrm{t}\right)$. Now the oscillator frequency is $\sqrt{\frac{2}{\mathrm{T}}}\sin \left({\omega }_{\mathrm{c}}\mathrm{t}+{45}^{\circ }\right)$

Using this, we calculate the vectors of demodulated symbols of  ${\mathrm{s}}_1\left(\mathrm{t}\right)$ and ${\mathrm{s}}_2\left(\mathrm{t}\right)$ along the basis function and get the following result.

${\mathrm{S}}_1=\sqrt{\frac{\mathrm{E}}{2}}$and ${\mathrm{S}}_2=-\sqrt{\frac{\mathrm{E}}{2}}$

Now, distance between vectors $\mathrm{d}=2\sqrt{\frac{\mathrm{E}}{2}}$

Now, using the relation that ${\mathrm{P}}_{\mathrm{e}}=\mathrm{Q}\left(\sqrt{\frac{{\mathrm{d}}^2}{2{\mathrm{N}}_{\mathrm{o}}}}\right)=\mathrm{Q}\left(\sqrt{\frac{4\times \frac{\mathrm{E}}{2}}{2{\mathrm{N}}_{\mathrm{o}}}}\right)=\mathrm{Q}\left(\sqrt{\frac{\mathrm{E}}{{\mathrm{N}}_{\mathrm{o}}}}\right)$

Alternate Solution:

For BPSK

For coherent detection when there is phase synchronization between the carrier and received signal:

$P_e=Q\left(\sqrt{\frac{2E_b}{N_o}}\right)$

For coherent detection when there is phase shift of ϕ  between the carrier and received signal:

$P_e=Q\left(\sqrt{\frac{2E_{b}Co{s}^2\varphi }{N_o}}\right)$

Substituting the values Cos² ϕ = 1/2

$P_e=Q\left(\sqrt{\frac{E}{N_o}}\right)$

When 1 is transmitted received signal is:

R = 1 + N

Probability of error $=\frac{1}{2}\left(1\right)\left(\frac{1}{4}\right)$

[Area of shaded region] = 1/8

When -1 is transmitted, the received signal is:

Probability of error = Area of the shaded region = 1/8

Total probability of error is:

P[E] = P(x = -1)P_E1 + P(x = 1) P_E2

$=\frac{1}{2}\left(\frac{1}{8}\right)+\frac{1}{2}\left(\frac{1}{8}\right)$

$=\frac{1}{8}=0.125$

Given:

Probability of error when  0 is received = p

Probability of error when  1 is received = 1 - p

If the number of zeros exceeds the number, the decoder decodes it as ‘0” otherwise as “ 1”. 

For n = 3 bits

Hence the probability of error is,

P_e = p³ + p²(1 - p) + p2(1 - p) + p2(1 - p) 

P_e = p³ + 3p2(1 - p)

$P_e=2Q\left[\sqrt{\frac{d_{min}^2}{2N_0}}\right]$

(Where 2 represents the number of decisions boundaries)

From the figure,

d_min = 0.5 (-0.25 to 0.25)

$\frac{N_0}{2}=\frac{1}{2}\Rightarrow N_0=1$

$P_e=2Q\left[\sqrt{\frac{{0.5}^2}{2\times 1}}\right]=2Q\left(\sqrt{0.125}\right)$

Probability of error is given by ${\mathrm{P}}_{\mathrm{e}}=Q\left(\sqrt{\frac{{\mathrm{d}}_{\mathrm{m}\mathrm{i}\mathrm{n}}^2}{2{\mathrm{N}}_{\mathrm{o}}}}\right)$

where ${\mathrm{d}}_{\mathrm{m}\mathrm{i}\mathrm{n}}$is distance between signaling points.

$\frac{{\mathrm{N}}_{\mathrm{o}}}{2}=0.5\times {10}^{-12}\Rightarrow {\mathrm{N}}_{\mathrm{o}}={10}^{-12}\frac{\mathrm{W}}{\mathrm{H}\mathrm{z}}$

Using ${\mathrm{P}}_{\mathrm{e}}=0.006$ and $\mathrm{Q}\left(2.5\right)=0.006$ 

we have, $\sqrt{\frac{{\mathrm{d}}_{\mathrm{m}\mathrm{i}\mathrm{n}}^2}{2{\mathrm{N}}_{\circ }}}=2.5$

$\Rightarrow {\mathrm{d}}_{\mathrm{m}\mathrm{i}\mathrm{n}}=\sqrt{({\left(2.5\right)}^2\times 2\times {\mathrm{N}}_{\mathrm{o}})}$

$\Rightarrow {\mathrm{d}}_{\mathrm{m}\mathrm{i}\mathrm{n}}=3.54\times {10}^{-6}\sqrt{\mathrm{J}\mathrm{o}\mathrm{u}\mathrm{l}\mathrm{e}}$

We have the probability of error ${\mathrm{P}}_{\mathrm{e}}=\mathrm{Q}\left(\sqrt{\frac{{\mathrm{d}}^2}{2\eta }}\right)$ where $\mathrm{d}$ is the distance between the constellation points.

For constellation $\mathrm{a},{\mathrm{d}}_{\mathrm{a}}^2={\left(2\mathrm{c}-\mathrm{c}\right)}^2+{\left(\mathrm{c}-\left(-2\mathrm{c}\right)\right)}^2=10{\mathrm{c}}^2$

For constellation $\mathrm{a},{\mathrm{d}}_{\mathrm{b}}^2={\left(\mathrm{A}-\left(-\mathrm{A}\right)\right)}^2=4{\mathrm{A}}^2$

For the two probabilities of error to be equal, ${\mathrm{d}}_{\mathrm{a}}^2={\mathrm{d}}_{\mathrm{b}}^2$

$\begin{array}{l}\Rightarrow 4A^2=10c^2\\ \Rightarrow \mathrm{A}=\frac{\sqrt{10}}{2}\mathrm{c}=1.58\end{array}$

Based on the information given for the two symbols waveforms, we see that the basis function is $\sqrt{\frac{2}{\mathrm{T}}}\sin \left({\omega }_{\mathrm{c}}\mathrm{t}\right)$. Now the oscillator frequency is $\sqrt{\frac{2}{\mathrm{T}}}\sin \left({\omega }_{\mathrm{c}}\mathrm{t}+{45}^{\circ }\right)$

Using this, we calculate the vectors of demodulated symbols of  ${\mathrm{s}}_1\left(\mathrm{t}\right)$ and ${\mathrm{s}}_2\left(\mathrm{t}\right)$ along the basis function and get the following result.

${\mathrm{S}}_1=\sqrt{\frac{\mathrm{E}}{2}}$and ${\mathrm{S}}_2=-\sqrt{\frac{\mathrm{E}}{2}}$

Now, distance between vectors $\mathrm{d}=2\sqrt{\frac{\mathrm{E}}{2}}$

Now, using the relation that ${\mathrm{P}}_{\mathrm{e}}=\mathrm{Q}\left(\sqrt{\frac{{\mathrm{d}}^2}{2{\mathrm{N}}_{\mathrm{o}}}}\right)=\mathrm{Q}\left(\sqrt{\frac{4\times \frac{\mathrm{E}}{2}}{2{\mathrm{N}}_{\mathrm{o}}}}\right)=\mathrm{Q}\left(\sqrt{\frac{\mathrm{E}}{{\mathrm{N}}_{\mathrm{o}}}}\right)$

Alternate Solution:

For BPSK

For coherent detection when there is phase synchronization between the carrier and received signal:

$P_e=Q\left(\sqrt{\frac{2E_b}{N_o}}\right)$

For coherent detection when there is phase shift of ϕ  between the carrier and received signal:

$P_e=Q\left(\sqrt{\frac{2E_{b}Co{s}^2\varphi }{N_o}}\right)$

Substituting the values Cos² ϕ = 1/2

$P_e=Q\left(\sqrt{\frac{E}{N_o}}\right)$