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Coded Excitation Signal Analysis
© 2010 Maplesoft
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Introduction
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In a scheme called coded excitation, a coded signal is used with a matched filter (correlator) to improve the signal to noise ratio without increasing the peak signal power. The setup is illustrated in Figure 1.
Figure 1 - A coded excitation scheme
In the above figure,  is the transmitted signal waveform,  is the impulse response of the transducers and propagation medium from source to receiver,  is the received signal, and  is the output of the correlator (matched filter). Accordingly,  , or
The output of the correlator is given by
Therefore  =  . Noting that the autocorrelation of a function  is given by  , therefore the output waveform can be expressed as  . Therefore, the more that the autocorrelation of the transmitted signal resembles a delta function, the better the ability to discern  . In other words, the sharper the autocorrelation, the better the axial resolution.
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Signal Design and Analysis
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In this section, we examine several different options for a coded excitation scheme by investigating the autocorrelation of several function. First we wish to define some intermediate functions that will be of use. The first is the offset function, which forms the interior of the autocorrelation integral. Note that the we are calculating the autocorrelation as it would occur in the correlator circuit with a delay between the two inputs, called the offset. This does not impact the final result.
The function multiplied by its time-offset is
The correlation is the time-integral of this function, and is given by
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Hard-Windowed FM Chirp
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In this example, the transmitted function of interest is an FM chirp with hard windowing. The window is simply a rectangular pulse starting at  and ending at  .
The FM chirp is a frequency-varying sinusoid multiplied by the rectangular window:
In the above, f0 is the center frequency and a is the rate of frequency increase. Choosing  Hz,   , and  gives
We may now examine the result of multiplying this function with an offset version of itself. In the animation below, the time offset is swept from -6 to +6 seconds, and the product of the two functions is shown in red. The autocorrelation will be the integral of these two functions for every possible offset.
The product at zero offset is 
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(3.1.1) |
We may view autocorrelation on a normalized dB scale:
This function is known symbolically and is 
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(3.1.2) |
The expression at zero offset is 
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(3.1.3) |
or numerically, 
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(3.1.4) |
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FM Chirp with Cosine Window
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Here we replace the step window with a cosine window to improve the dropoff of the sidelobes at the cost of a wider mainlobe. This time the window again starts at  and ends at  , and has a parameter  which is the ratio of tapered to constant segments.
The FM chirp is a frequency-varying sinusoid multiplied by the cos window:
Choosing the previous parameters and including  gives
We may now examine the result of multiplying this function with an offset version of itself. In the animation below, the time offset is swept from -6 to +6 seconds, and the product of the two functions is shown in red. The autocorrelation will be the integral of these two functions for every possible offset.
We may view compare autocorrelation on a normalized dB scale:
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Gaussian-Modulated Sinusoid
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Now we consider a Gaussian-modulated sinusoid.
Choosing  Hz and  Hz gives
We may now examine the result of multiplying this function with an offset version of itself. In the animation below, the time offset is swept from -4 to +4 seconds, and the product of the two functions is shown in red. The autocorrelation will be the integral of these two functions for every possible offset.
We may view compare autocorrelation on a normalized dB scale:
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