# How to calculate auto-correlation of a bpsk modulated signal or how to calculate expectation value of complex exponential function

How to calculate auto-correlation of a bpsk modulated signal, or how to calculate expectation value of complex exponential function manually not by using matlab or any other software?

For example, if

$$x(t)= a\exp(i2\pi ft)+n(t)$$

how would you solve

$$\mathrm{E}[x(t)]=?$$

and

$$R(t,t_1)= \mathrm{E}[x(t+t_1/2)x'(t-t_1/2)]=?$$

where $a$ is either $+1$ or $-1$ mapped from the binary representation of the ASCII value of any text message.

• For some basic information about writing math at this site see e.g. here, here, here and here. – Kasper Apr 26 '13 at 15:04
• This is likely to be a better fit on the signal processing stackexchange dsp.SE. – Dilip Sarwate Apr 26 '13 at 15:09
• I agree with @DilipSarwate. DSP.SE is a better fit. – Peter K. Apr 26 '13 at 16:19
• Neha, your question was automatically migrated here. No need to duplicate post. I have deleted the duplicate. – Peter K. Apr 26 '13 at 16:53
• Which do you want to know- auto-correlation or expected value? The two are totally different things. – Jim Clay Apr 26 '13 at 18:13

What is the autocorrelation function of the random process $$X(t) = A\exp(j2\pi f_ct), -\infty < t < \infty$$ where $A$ is a random variable equally likely to have values $+1$ and $-1$, and the autocorrelation function is defined as $$R_X(t,\tau) = E[X(t+\tau/2)X^*(t-\tau/2)], -\infty < t,\tau < \infty.$$
This is easy since the only random variable involved is $A$, and it is a discrete random variable. Thus, we have \begin{align} E[X(t)]&= E[A]\cdot \exp(j2\pi f_ct)\\ &= 0 ~~\scriptstyle{\text{since }} E[A]=0.\\ R_X(t,\tau) &= E[X(t+\tau/2)X^*(t-\tau/2)],\\ &= E[A^2]\cdot \exp\left(j2\pi f_c(t+\tau/2)-j2\pi f_c(t-\tau/2)\right)\\ &= \exp(j2\pi f_c\tau)~~\scriptstyle{\text{since }} E[A^2]=1. \end{align}
What would be needed to make this question worth answering? First, one usually has a baseband pulse shape, usually time-limited, that is nowhere mentioned by the OP. The autocorrelation function of a BPSK signal is usually a scaled version of the autocorrelation of the baseband pulse shape (and so the power spectral density is determined by the shape of the baseband pulse). To get to this result, one needs to consider not just one bit but rather the stream of bits that modulate the BPSK signal, and assume that these bits are i.i.d. random variables (such as $A$ above). Second, something needs to be said about the noise $n(t)$ that the OP casually includes in her question.