Chris Smemoe , Computer
Science 450
Table of Contents 1. Rect function * Rect function
2. Rect function * Gaussian function
3. Gaussian function * Gaussian function
4. Rect function * Noise function
5. Gaussian function * Noise function
Convolve the Rectangular Pulse (file: 1D_Rect.dat) with itself and plot the result. (2 pts.) (Sketch the result or include a copy of the plot in your write-up.)
Explain in your write-up why you get this result. (1 pt.)
You get the triangular pulse for the result because the convolution of two rectangular functions is a triangular pulse. The two rectangular pulses "smooth" each other to produce the triangular pulse.
If we consider one of these pulses the "signal" and the other the "filter," what kind of filtering (Section 9.5, of your text) is going on? (2 pts.)
It appears as if one rectangular function is smoothing the other function. Basically, the rectangular pulse moves along the other rectangular pulse, producing a local average of the other rectangular function over a unit width interval.
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Convolve 1D_Rect.dat with a Gaussian (file: 1D_Gauss.dat) and plot the result. (2 pts.) (Sketch the result or include a copy of the plot in your write-up.)
Explain in your write-up why you get this result. (1 pt.)
Because the Gaussian has the effect of "smoothing" the function it comes in contact with. In this case, since it came in contact with a square function, it rounded it out along the edges.
If we consider 1D_Rect.dat the signal and 1D_Gauss.dat the filter, what kind of filtering (Section 9.5 of your text) is going on? (2 pts.)
A smoothing filter.
How does the effect of this filter compare with what is happening in A.3 (above)? (2 pts.)
In A.3, two rectangular pulses are convoluted. This always produces a triangular pulse. In this example, a rectangular pulse is convolved with a Gaussian. When the Gaussian is convolved with the rectangular function, the original function becomes more smoothed along the entire length of the function and looks less like a triangle.
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Convolve a Gaussian with itself (file: 1D_Gauss.dat) and plot the result. (2 pts.) (Sketch the result or include a copy of the plot in your write-up.)
Explain MATHEMATICALLY in your write-up why you get this result. (3 pts.)
The convolution of two Gaussian functions always produces another Gaussian, according to section 9.4.3 in the book. In this section, the following equation is given:
where c = a+b and 
.
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Convolve 1D_Rect.dat with a noisy signal (file: 1D_Noise.dat) and plot the result. (2 pts.) (Sketch the result or include a copy of the plot in your write-up.)
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Convolve 1D_Gauss.dat with a noisy signal (file: 1D_Noise.dat) and plot the result. (2 pts.) (Sketch the result or include a copy of the plot in your write-up.)
Compare and contrast the result from D.1 with that from E.1. Which has the greatest effect on 1D_Noise.dat? Why? (2 pts.)
You can see the original noise function above. Apparently, the rectangular function has the greatest effect of reducing the noise on the original noise function since the rectangular function averages a wider range of values over the noise function. The Gaussian tends to do more of a "local smoothing" operation on the noise function.
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Implement the (1D) edge enhancement described in Section 9.5.2 using 1D_Gauss.dat and f(x) = file: 1D_Edge.dat. Sketch the result or include a copy of the plot in your write-up. (2 pts.)
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smemoe@byu.edu
Date last modified: 10/12/1999