0.7 Bits to symbols to signals (Page 8/8)

Software receiver design Page 8 / 8

- 1, - 1, 1, 1, 1, 3, 5, 7, 7, 7, 5, 5 .

For marker B, starting at the same point in the data sequence and performing the associated moving weighted sum, produces

1, 1, 3, - 1, - 5, - 1, - 1, 1, 7, - 1, 1, - 3 .

With the two correlator output sequences shown, started two values prior to the start of the seven-symbol marker, we want theflag indicating a frame start to occur with point number 9 in the correlator sequences shown.Clearly, the correlator output for marker B has a much sharper peak at its ninthvalue than the correlator output of marker A. This should enhance the robustness of the use of marker Brelative to that of marker A against the unavoidable presence of noise.

The binary signal from Equation 8 is shown with a seven symbol embedded marker m_1, m_2, ..., m_7. The process of correlation multiples successive elements of the signal by the marker, and sums. The marker shifts to the right and the process repeats. The sum is largest when the marker is aligned with itself because all the terms are positive, as shown in the bottom part of the diagram. Thus the shift with the largest correlation sum points towards the embedded marker in the data sequence. Correlation can often be useful even when the data is noisy. — The binary signal from [link] is shown with a seven symbol embedded marker $m_{1}$ , $m_{2}$ , $. . .$ , $m_{7}$ . The process of correlation multiples successive elementsof the signal by the marker, and sums. The marker shifts to the right and the process repeats. The sum is largest when the marker is aligned with itselfbecause all the terms are positive, as shown in the bottom part of the diagram. Thus the shift with the largest correlation sum pointstowards the embedded marker in the data sequence. Correlation can often be useful even when the data is noisy.

Marker B is a “maximum-length pseudonoise (PN)” sequence. One property of a maximum-length PN sequence ${c_{i}}$ of plus and minus ones is that its autocorrelation is quite peaked:

R_{c} (k) = \frac{1}{N} \sum_{n = 0}^{N - 1} c_{n} c_{n + k} = \{\begin{matrix} 1, & k = ℓ N \\ \frac{- 1}{N}, & k \neq ℓ N \end{matrix} .)

Another technique that involves the chunking of data and the need to locate boundaries between chunksis called scrambling . Scrambling is used to “whiten” a message sequence(to make its spectrum flatter) by decorrelating the message.The transmitter and receiver agree on a binary scrambling sequence $s$ that is repeated over and over to form a periodic string $S$ that is the same size as the message. $S$ is then added (using modulo 2 arithmetic) bit by bit to the message $m$ at the transmitter, and then $S$ is added bit by bit again at the receiver.Since both $1 + 1 = 0$ and $0 + 0 = 0$ ,

m + S + S = m

and the message is recaptured after the two summing operations.The scrambling sequence must be aligned so that the additions at the receiver correspond to theappropriate additions at the transmitter. The alignment can be accomplishedusing correlation.

Redo the example of this section, using M atlab .

Add a channel with impulse response $1, 0, 0, a, 0, 0, 0, b$ to this example. (Convolve the impulse response of the channel with the data sequence.)

For $a = 0.1$ and $b = 0.4$ , how does the channel change the likelihood that the correlation correctly locatesthe marker? Try using both markers $A$ and $B$ .
Answer the same question for $a = 0.5$ and $b = 0.9$ .
Answer the same question for $a = 1.2$ and $b = 0.4$ .

Generate a long sequence of binary random data with the marker embedded every 25 points. Check that marker Ais less robust (on average) than marker B by counting the number of times marker A misses the frame start comparedwith the number of times marker B misses the frame start.

Create your own marker sequence, and repeat the previous problem. Can you find one that does better thanmarker B?

Use the 4-PAM alphabet with symbols $\pm 1, \pm 3$ . Create a marker sequence, and embed it in a long sequenceof random 4-PAM data. Check to make sure it is possible to correctly locate the markers.

Add a channel with impulse response $1, 0, 0, a, 0, 0, 0, b$ to this 4-PAM example.

For $a = 0.1$ and $b = 0.4$ , how does the channel change the likelihood that the correlation correctly locatesthe marker?
Answer the same question for $a = 0.5$ and $b = 0.9$ .

Choose a binary scrambling sequence $s$ that is 17 bits long. Create a message that is 170 bits long, andscramble it using bit-by-bit mod 2 addition.

Assuming the receiver knows where the scrambling begins, add $s$ to the scrambled data and verify that the output is the same as the original message.
Embed a marker sequence in your message. Use correlation to find the marker and toautomatically align the start of the scrambling.

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Source: OpenStax, Software receiver design. OpenStax CNX. Aug 13, 2013 Download for free at http://cnx.org/content/col11510/1.3

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