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Binary sequence generated by shift register with feedback have periodic properties. A simple 4-bit shift register shown in Fig (a). For the initial condition shown, it can be verified that the output sequence is 000100110101111. The sequence repeats itself after 15 bits. Let binary one be 1 V and binary zero be -1 V, and let the voltage level be held constant during the bit interval. The resulting time waveform is shown in Fig (b), and its time autocorrelation function is shown in Fig. (c). This is my problem, I did not understand the following relationships $\rho(\tau)=-\frac{1}{15}$ $\rho(0)=1$ and the Fig. (c).

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Do you know how the autocorrelation function is defined? What part of applying the definition are you finding difficult? –  joriki Mar 29 '12 at 6:55
    
$\rho(\tau)=\frac{1}{15}\sum_{n=1}^{15}x(n)x(n+\tau)$ This? $\rho(0)=\frac{1}{15}\sum_{n=1}^{15}x^{2}(n)=1$ ok; $\rho(\tau)=-1/15$ because number_of(0)=number_of(1)+1? –  Mark Mar 29 '12 at 7:32
    
I suggest you take a look at the Wikipedia article. There are two different ways of defining the autocorrelation function; one subtracts the mean and normalizes by the variance and the other doesn't. It seems that the image you include uses the former definition, whereas you're quoting the latter. –  joriki Mar 29 '12 at 8:52
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