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Quadrilateral ABCD M is midpoint of diagonal AC. To Prove: Area of quadrilateral ABMD equals area of quadrilateral CBMD. Plan: Use Principle 3 to obtain two pairs of triangles which are equal in area. Then use the Addition Postulate.

lo( 5

PROOF:

which is not zero for w, p l 1 5 n. Therefore, there is no value for w, < n , and no input x(n) that will generate o the given output y(n).

Given:

Let h(n) be the unit sample response of an ideal low-pass filter with acutoff frequency wc = n/4. Shown in the figure below is a linear shift-invariant system that is formed from a cascade of a low-pass filter and two modulators. Find the frequency response of the overall system relating the input x(n) to the output y(n).

Dx (x 2 y3 5xy2 4y) = Dx (4) x 2 (3y2 y ) + y3 (2x) 5(x(2yy ) + y2 ) 4y = 0 3x 2 y2 y + 2xy3 10xyy 5y2 4y = 0 Substitute 3 for x and 2 for y, 108y + 48 60y 20 4y = 0 44y + 28 = 0 y =

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Statements 1. M is the midpoint of AC. 2. BM is a median of ^ACB. DM is a median of ^ACD. 3. Area (^AMB) area(^BMC), Area(^AMD) area(^DMC). 4. Area of quadrilateral ABMD equals area of quadrilateral CBMD.

There are two ways that we may use to find the frequency response of this system. The first is to note that because the input to the low-pass filter is (- I)"x(n), the output of the low-pass filter is

Reasons 1. Given 2. A line from a vertex of a triangle to the midpoint of the opposite side is a median. 3. A median divides a triangle into two triangles of equal area. 4. If equals are added to equals, the results are equal.

Therefore. Bringing the term (-1)" inside the summation, and using the fact that ( - I ) " - ~ = (we have

Another way to determine the frequency response is to find the response of the system to a complex exponential, x(n) = eln'". Modulating by (-1)" = e-inn produces the sequence

The areas of similar polygons are to each other as the squares of any two corresponding segments. Thus if ^ABC ~ ^A B C and the area of ^ABC is 25 times the area of ^A B C , then the ratio of the lengths of any two corresponding sides, medians, altitudes, radii of inscribed or circumscribed circles, and such is 5:1.

7 28 = 44 11

which is the input to the LSI system. Because u(n) is a complex exponential, the response of the system to v(n) is

9.11 Ratio of areas and segments of similar triangles Find the ratio of the areas of two similar triangles (a) if the ratio of the lengths of two corresponding sides is 3:5; (b) if their perimeters are 12 and 7. Find the ratio of the lengths of a pair of (c) corresponding sides if the ratio of the areas is 4:9; (d ) corresponding medians if the areas are 250 and 10.

In our study of discrete-time signals and systems we will be concerned with the manipulation of signals. These manipulations are generally compositions of a few basic signal transformations. These transformations may be classified either as those that are transformations of the independent variable n or those that are transformations of the amplitude of x ( n ) (i.e., the dependent variable). In the following two subsections we will look briefly at these two classes of transformations and list those that are most commonly found in applications.

9 25 144 49

Sequences are often altered and manipulated by modifying the index n as follows:

2 3 5

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