[Solved] Project 3: Convolution (Matched Filtering)

Introduction

The acquisition of a signal in a digital communications system requires the convergence of several signal processing technologies. The major receiver technologies are (1) ADCs, (2) timing recovery, (3) carrier recovery, (4) channel equalization, (5) signal detection, and (6) matched filtering (convolution).

Matched filtering

Practical receivers often estimate the transmitted signal by using a technique known as matched filtering. A receiver employing such a strategy possesses filters the received signal with a filter whose shape is matched to the transmitted signals pulse shape. Specifically the matched filters pulse shape is a time-reversed version of the transmit pulse shape. Thus, if the transmitted pulse shape h(t) is known to be:

h(t) for 0 t T

then the matched filters response h_m(t) is:

h_m(t) = h(Tt) for 0 t T

Such processing provides two advantages. By filtering the received signal with a matched filter, only thoes frequencies contained in the transmitted signals spectrum are process while the remaining frequencies are ignored or nulled. Therefore matched filters limit the amount of the noise that is passed on to subsequent receiver stages, maximizing the receivers SNR. A second advantage is that a matched filter correlates the received signal with the transmit pulse shape over the symbol period T. Such processing results in a correlation gain by integrating the received signal energy while averaging out the zero-mean AWGN.

An example of matched filtering is shown in Figure 1.

1

Figure 1: Signal with and without noise (left) and their matched filter outcomes (right) measured at the end of each message period.

Spread Spectrum (Mitra)

Consider a multiuser spread spectrum communication system in which the transmitted data sequence is obtained by modulation a low-rate set of data symbols with a high-rate spreading sequence, typically chosen to be periodic pseudo-random sequence or orthogonal basis set (e,g,; Walsh functions). The elements of the spreading sequence are called chips. The number of chips per symbol is called the processing gain (denoted N). The following N=5 example illustrates how the system works.

The data (symbols) to be transmitted is m[k] = { -1 -1 +1 +1 -1}

For each of 4-distinct users a binary-valued random spreading sequence, denoted s_i (see below) is defined providing a processing gain N=5.

s₁: { -1 +1 -1 +1 +1}: user 1

s₂: { -1 -1 -1 -1 +1}: user 2

s₃: { -1 +1 -1 -1 +1}: user 3

s₄: { +1 -1 -1 -1 +1}: user 4

Generate an up-sampled N=5 data set of 5 symbols (note MATLAB indexing begins indexing with 1 not 0) based on m[k].

d[k]

-1

-1

-1

1

1

Figure 2: Up-sampled (by 5) slow-rate data sequence m[k] to produce d[k]=[-1 -1 -1 -1 -1: -1 -1 -1 -1 -1: 1 1 1 1 1: 1 1 1 1 1: -1 -1 -1 -1 -1 ] for N=5 (n[0,24]).

Suppose a message is intended for user 1. The transmitter generates a high-rate spreading sequence S₁[k] based on s₁[k].

s1[k]

s1[k]

s1[k]

s1[k]

s1[k]

S1[k]

Figure 3: Periodic spreading sequence S₁[k]=[-1 1 -1 1 1: -1 1 -1 1 1: -1 1 -1 1 : -1 1 -1 1 1: -1 1 -1 1 1 ] for user 1 (N=5) (n[0,24]) based on s₁[k] = [-1 1 -1 1 1].

Modulate the up sampled d[k] with S₁[k] (i.e., d[k]*S₁[k] ) using point-by-point multiplicative modulation.

Figure 4: Product modulated sequence p[k]= d[k]*S₁[k] =[1 -1 1 -1 -1: 1 -1 1 -1 -1: -1 1 -1 1 1: -1 1 -1 1 1: 1 -1 1 -1 -1] for intended user 1(N=5) (n[0,24]) (i.e., d[k]*S₁[k] )

Define the digital matched filter as the reverse image of s₁[k] = [-1 1 -1 1 1], namely h_1m[k] = [1 1 -1 1 -1 ]. That is, the matched filter is the reverse image of s₁. Convolve the received signal with the matched filter h_1m[k].

-1

1

1

-1

-1

Figure 5:Output of the matched filter: y₁[k] = [ 1 0 -1 2 -5: 3 -1 -1 3 -5: 1 -1 1 -1 5: -3 1 1 -3 5: -1 1 -1 1 -5] for user 1; (N=5) (n[0,24])

Notice that at the end of each 5sample periods (k=5, 10, 15, 20, 25), the matched filter achieves its maximal output value of 5 corresponding to a processor gain of N=5. Using a decision threshold 4, for example, the original message m[k] = {-1 -1 +1 +1 -1} is recovered without error. With noise present, a more careful threshold selection would be needed.

Project 3

Project 3 is an extension of the spread-spectrum N=5 example considered in the previous section.

Generate, using MATLAB:

• a 20 random symbol (1 valued) low-rate data message m[k]. For N=5, the transmitted signal d[k] consists of 100 samples.
• a user #1 N=5 random spreading sequence S₁[k]. That is, copy s₁[k] 20 times to create a periodic 100-sample signal S₁[k].
• a 100 sample product modulated signal p[k].

Perform the modulation and match filtering as practiced in the opening example.

Q1: Using d[k] and S₁[k] (message directed to user #1), generate the matched filter output and determine the bit error rate (BER) for a rationale choice of decision threshold.

Q2: Repeat Q1 except use the matched filter for user #2. Determine the bit error rate (BER) for a rationale choice of decision threshold.

Q3: Add normal noise to the message d[k] for a SNR level of SNR = 0dB verify. Repeat the Q1 study and report. Determine the bit error rate (BER) for a rationale choice of decision threshold.

Due Oct.Oct.31, 2016

Same format as last project.