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12.2 Coordinates of points in the four quadrants What are the signs of the coordinates of (a) a point in quadrant I; (b) a point in quadrant II; (c) a point in quadrant III; (d) a point in quadrant IV Show which coordinate has a sign and which zero value for a point between quadrants (e) IV and I; (f) I and II; (g) II and III; (h) III and IV.

This system is called an ideal discrete-to-continuous (DIC) converter. Because the impulse response of the reconstruction filter is

Thus, in Fig. 1-41(a) the angles of a and b are adjacent supplementary angles. However, in Fig. 1-41(b) the supb 180 . Either of two supplementary angles is said to be plementary angles are nonadjacent. In each case, a the supplement of the other.

Fig. 3-6. (a) A discrete-to-continuous converter with an ideal low-pass reconstruction filter. (h) The frequency response of the ideal reconstruction filter.

PRINCIPLE 1:

xa(t) =

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1: If f (x) > 0 for all x in (a, b), then the graph of f is concave upward between x = a and x = b If f (x) < 0 for all x in (a, b), then the graph of f is concave downward between x = a and x = b For a rigorous proof of Theorem 231, see Problem 2317 Corollary 232: If the graph of f has an in ection point at x = c, and f exists and is continuous at x = c, then f (c) = 0 In fact, if f (c) = 0, then f (c) > 0 or f (c) < 0.

Fig. 1-42

n=-00

PRINCIPLE 2:

This interpolation formula shows how x,(t) is reconstructed from its samples x(n) = x,(nTs). In the frequency domain, the interpolation formula becomes

T ~ X ( ~ J * ~ S )

Fig. 1-43

If f (c) > 0, then f (x) > 0 for all x in some open interval containing c, and the graph would be concave upward in that interval, contradicting the assumption that there is an in ection point at x = c We get a similar contradiction if f (c) < 0, for in that case, the graph would be concave downward in an open interval containing c EXAMPLES.

Thus, x ( e i w )is frequency scaled (o= QTS), and then the low-pass filter removes all frequencies in the periodic above the cutoff frequency Q. = TIT,. , spectrum x(eiQTr) Because it is not possible to implement an ideal low-pass filter, many D/A converters use a zero-order hold for the reconstruction filter. The impulse response of a zero-order hold is

Thus if AB and CD are straight lines in Fig. 1-43, then /1 > /3 and /2 > /4. Hence, if m/1 40 ; in such a case, m/2 m/4 140 .

OitlT,

PRINCIPLE 3:

ho(0 =

40 , then m/3

After a sequence of samples xa(nT,)has been converted to impulses, the zero-order hold produces the staircase approximation to xu(!)shown in Fig.3-7. With a zero-order hold, it is common to postprocess the output with a reconstruction compensation filter that approximates the frequency response

(a) Consider the graph of y = x 3 [see Fig. 23-4(a)]. Here y = 3x 2 and y = 6x. Since y > 0 when x > 0, and y < 0 when x < 0, the curve is concave upward when x > 0, and concave downward when x < 0. There is an in ection point at the origin, where the concavity changes. This is the only possible in ection point, for if y = 6x = 0, then x must be 0.

2. T

PRINCIPLE 4:

3. T

50 [Fig. 1-44(a) or (b)].

4. T

Fig. 1-44

(b) If f (c) = 0, the graph of f need not have an in ection point at x = c. For instance, the graph of f (x) = x 4 [see Fig. 23-4(b)] has a relative minimum, not an in ection point, at x = 0, where f (x) = 12x 2 = 0.

2. T

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