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then which is the unit sample response of a low-pass filter with a magnitude of and a cutoff frequency w = n / 2 . The second way to work this problem is to find the frequency response of the system that has a unit sample response g(n) = h(2n), given that H(el'") is the frequency response of a system with a unit sample response h(n). Although more difficult than the first approach. this will give a general expression for the frequency response G(eJw)in terms of H(ejw)that may be applied to any system. To find the frequency response, we must evaluate the sum

2 2 2

I +(-I)" =

(9.7)

With ~ ( e jthe frequency response of a low-pass filter with a cutoff frequency w, = n/4, this gives the same result ~ ) as before.

Euclid s Elements began with only five postulates. We shall go through them with an eye for alternatives.

21 .0

POSTULATE 1:

Consider the high-pass filter that has a cutoff frequency w, = 3n/4as shown in the following figure:

2.3 Prove by use of coordinates that the midpoint of the hypotenuse of a right triangle is equidistant from the three vertices.

One and only one straight line can be drawn through any two points. ( 2, Postulate 11)

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(a) Find the unit sample response, h(n). (h) A new system is defined so that its unit sample response is h l ( n ) = h(2n). Sketch the frequency ( response, H Ie j " ) , of this system.

We further postulate that there exist at least two points, A and B, so there exists a straight line.

(a) The unit sample response may be found two different ways. The first is to use the inverse DTFT formula and

H ( e J u ) may be written as H ( @ ) = HIp(eJ("-"' 1

POSTULATE 2:

Let the origin of a coordinate system be located at the right angle C; let the positive x-axis contain leg CA and the positive y-axis leg CB [see Fig. 2-7(a)]. Vertex A has coordinates (b, 0), where b = CA; and vertex B has coordinates (0, a), where a = BC. Let M be the midpoint of the hypotenuse. By the midpoint formulas (2.2), the coordinates of M are (b/2, a/2).

H I p ( e J W= )

A line segment can be extended in either direction indefinitely. ( 1, description of a line)

h ( n ) = eJn"hlp(n) ( - l ) " h l p ( n ) =

Traditionally, we suppose that the line through A and B looks like Fig. 19-1(a), but our postulates do not rule out the possibility of Fig. 19-1(b). In Fig. 19-1(b), the line reaches no end in either direction, and thus in some sense can be extended indefinitely. Remember that the original meaning of geometry comes from earth and measure. The straightest line that could be drawn on the earth would wrap around to form a great circle, as discussed in 17.

(6) The frequency response of the system that has a unit sample response h I ( n ) = h ( 2 n ) may be found by evaluating the discrete-time Fourier transform sum directly:

Fig. 19-1

n=-00

Fig. 2-7

is which is a low-pass filter with a cutoff frequency of n / 2 and a gain of f . A plot of H , ( ~ J " )shown in the following figure:

A circle can be drawn with any center and radius. ( 2, Postulate 14)

The ideal filters that have frequency responses as shown in the figure below are connected in cascade.

POSTULATE 3:

For an arbitrary input x(n), find the range of frequencies that can be present in the output y(n). Repeat for the case in which the two systems are connected in parallel.

CHAP. 2]

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