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algebra ( 1)n 1 will be 1 when n is odd and 1 when n is even. n! stands for the product 1 2 3 n of the rst n positive integers.

sin(nwl) . + hz(n) where hl(n) = --- 1s an ideal low-pass filter with a cutoff frequency of wl, and

Fig. 11-4

h2(n) = &(n)-

The locus of points equidistant from two given intersecting lines is the bisectors of the angles formed by the lines (Fig. 11-5).

A bandpass filter with a lower cutoff frequency of wl = 3n/8, an upper cutoff frequency of wz = 5 ~ / 8 and a gain , of one half.

Fig. 11-5

A high-pass filter with a cutoff frequency w, = ~ / 3 .

Differentiation of (0), using the power chain rule for Dx y3 and the product rule for Dx (xy), gives 3y2 y (xy + y) = 0 3y2 y xy y = 0 (3y2 x)y y = 0 Next, differentiate (1), (3y2 x)Dx y + y Dx (3y2 x) y = 0 (3y2 x)y + y (6yy 1) y = 0 (3y2 x)y + y ((6yy 1) 1) = 0 (3y2 x)y + y (6yy 2) = 0 [by the product rule] [by the power chain rule] [factor y ] (2) (1)

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FOURIER ANALYSIS 2n 3 n 5n ( b ) - < lo1 < -. 6 6

PRINCIPLE 5:

lo1 c -.

Fig. 11-6

2n 3

CHAP. 22]

The locus of points equidistant from two concentric circles is the circle concentric with the given circles and midway between them (Fig. 11-6). The locus of points at a given distance from a given point is a circle whose center is the given point and whose radius is the given distance (Fig. 11-7).

Unique, h ( n ) = ;6(n).

PRINCIPLE 6:

a for lo1 < n and it decreases linearly to zero at w = -, 4

zt-.

Fig. 11-7

Now solve (1) for y , y = Finally, substitute into (2) and solve for y , (3y2 x)y + (3y2 x)3 y + (3y2 x) y 3y2 x y 3y2 x 6y2 2 =0 3y2 x 6y2 2 =0 3y2 x multiply by (3y2 x)2 a b c = b ca a y 3y2 x

y(n) =

PRINCIPLE 7:

. .,

Fig. 11-8

i, - i, -2. i. g, i, -I].

The locus of points at a given distance from a given line is a pair of lines, parallel to the given line and at the given distance from the given line (Fig. 11-8). The locus of points at a given distance from a given circle whose radius is greater than that distance is a pair of concentric circles, one on either side of the given circle and at the given distance from it (Fig. 11-9). The locus of points at a given distance from a given circle whose radius is less than the distance is a circle, outside the given circle and concentric with it (Fig. 11-10). (If r d, the locus also includes the center of the given circle.)

3

(3y2 x)

PRINCIPLE 8:

Most discrete-time signals come from sampling a continuous-time signal, such as speech and audio signals, radar and sonar data, and seismic and biological signals. The process of converting these signals into digital form is called analog-to-digital (AID) conversion. The reverse process of reconstructing an analog signal from its samples is known as digital-to-analog ( D / A )conversion. This chapter examines the issues related to A/D and D/A conversion. Fundamental to this discussion is the sampling theorem, which gives precise conditions under which an analog signal may be uniquely represented in terms of its samples.

An A/D converter transforms an analog signal into a digital sequence. The input to the A/D converter, x,(t),

PRINCIPLE 9:

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