SUPPRESSION OF NARROWBAND INTERFERENCE IN A WIDEBAND SIGNAL

11.3 SUPPRESSION OF NARROWBAND INTERFERENCE IN A WIDEBAND SIGNAL

Let us assume that we have a signal sequence {x(n)} that consists of

a desired wideband signal sequence, say {w(n)}, corrupted by an ad- ditive narrowband interference sequence {s(n)}. The two sequences are uncorrelated. This problem arises in digital communications and in signal

Suppression of Narrowband Interference in a Wideband Signal 601

detection, where the desired signal sequence {w(n)} is a spread-spectrum signal, while the narrowband interference represents a signal from another user of the frequency band or some intentional interference from a jammer who is trying to disrupt the communication or detection system.

From a filtering point of view, our objective is to design a filter that suppresses the narrowband interference. In effect, such a filter should place

a notch in the frequency band occupied by the interference. In practice, however, the frequency band of the interference might be unknown. More- over, the frequency band of the interference may vary slowly in time.

The narrowband characteristics of the interference allow us to esti- mate s(n) from past samples of the sequence x(n) = s(n) + w(n) and to subtract the estimate from x(n). Since the bandwidth of {s(n)} is nar- row compared to the bandwidth of {w(n)}, the samples of {s(n)} are highly correlated. On the other hand, the wideband sequence {w(n)} has

a relatively narrow correlation.

The general configuration of the interference suppression system is shown in Figure 11.3. The signal x(n) is delayed by D samples, where the delay D is chosen sufficiently large so that the wideband signal com- ponents w(n) and w(n − D), which are contained in x(n) and x(n − D), respectively, are uncorrelated. The output of the adaptive FIR filter is the estimate

N −1 !

s(n) = ˆ

h(k)x(n − k − D)

k=0

The error signal that is used in optimizing the FIR filter coefficients is e(n) = x(n) − ˆ s(n). The minimization of the sum of squared errors again leads to a set of linear equations for determining the optimum coefficients. Due to the delay D, the LMS algorithm for adjusting the coefficients recursively becomes

k = 0, 1, . . . , N − 1

h n (k) = h n−1 (k) + △e(n)x(n − k − D),

n = 1, 2, . . .

FIGURE 11.3 Adaptive filter for estimating and suppressing a narrowband in- terference

Chapter 11

APPLICATIONS IN ADAPTIVE FILTERING

FIGURE 11.4 Configuration of modules for experiment on interference suppres- sion

11.3.1 PROJECT 11.2: SUPPRESSION OF SINUSOIDAL INTERFERENCE Three basic modules are required to perform this project.

1. A noise signal generator module that generates a wideband sequence {w(n)} of random numbers with zero mean value. In particular, we may generate a sequence of uniformly distributed random numbers using the rand function as previously described in the project on system identification. The signal power is denoted as P w .

2. A sinusoidal signal generator module that generates a sine wave se- quence s(n) = A sin ω 0 n, where 0 < ω 0 < π and A is the signal ampli- tude. The power of the sinusoidal sequence is denoted as P s .

3. An adaptive FIR filter module using the lms function, where the FIR filter has N tap coefficients that are adjusted by the LMS algorithm. The length N of the filter is an input variable to the program.

The three modules are configured as shown in Figure 11.4. In this project the delay D = 1 is sufficient, since the sequence {w(n)} is a white noise (spectrally flat or uncorrelated) sequence. The objective is to adapt the FIR filter coefficients and then to investigate the characteristics of the adaptive filter.

It is interesting to select the interference signal to be much stronger than the desired signal w(n), for example, P s = 10P w . Note that the power P x required in selecting the step size parameter in the LMS algo- rithm is P x =P s +P w . The frequency response characteristic H(e jω ) of the adaptive FIR filter with coefficients {h(k)} should exhibit a resonant peak at the frequency of the interference. The frequency response of the interference suppression filter is H s (e jω ) = 1 − H(e jω ), which should then exhibit a notch at the frequency of the interference.

It is interesting to plot the sequences {w(n)}, {s(n)}, and {x(n)}. It is also interesting to plot the frequency responses H(e jω ) and H s (e jω ) after the LMS algorithm has converged. The short-time average squared error ASE(m), defined by (11.12), may be used to monitor the conver- gence characteristics of the LMS algorithm. The effect of the length of the adaptive filter on the quality of the estimate should be investigated.

Adaptive Channel Equalization 603

The project may be generalized by adding a second sinusoid of a different frequency. Then H(e jω ) should exhibit two resonant peaks, pro-

vided the frequencies are sufficiently separated. Investigate the effect of the filter length N on the resolution of two closely spaced sinusoids.