Exercise 3.74 · 习题

Exercise 3.74: Alyssa P. Hacker is designing a
system to process signals coming from physical sensors. One important feature
she wishes to produce is a signal that describes the
zero crossings
of the input signal. That is, the resulting signal should be +
1 whenever the
input signal changes from negative to positive, −
1 whenever the input signal
changes from positive to negative, and 0 otherwise. (Assume that the sign of a
0 input is positive.) For example, a typical input signal with its associated
zero-crossing signal would be

… 1 2 1.5 1 0.5 -0.1 -2 -3 -2 -0.5 0.2 3 4 …
… 0 0 0 0 0 -1 0 0 0 0 1 0 0 …

In Alyssa’s system, the signal from the sensor is represented as a stream
sense-data and the stream zero-crossings is the corresponding
stream of zero crossings. Alyssa first writes a procedure
sign-change-detector that takes two values as arguments and compares the
signs of the values to produce an appropriate 0, 1, or −
1. She then
constructs her zero-crossing stream as follows:

(define (make-zero-crossings
input-stream last-value)
(cons-stream
(sign-change-detector
(stream-car input-stream)
last-value)
(make-zero-crossings
(stream-cdr input-stream)
(stream-car input-stream))))

(define zero-crossings
(make-zero-crossings sense-data 0))

Alyssa’s boss, Eva Lu Ator, walks by and suggests that this program is
approximately equivalent to the following one, which uses the generalized
version of stream-map from Exercise 3.50:

(define zero-crossings
(stream-map sign-change-detector
sense-data
⟨expression⟩))

Complete the program by supplying the indicated ⟨expression⟩.

练习 3.74：Alyssa P. Hacker 正在设计一个处理物理传感器信号的系统。她希望实现的一个重要特性是生成描述输入信号零交叉点 (zero crossings) 的信号：当输入信号从负变正时，结果信号应为 +1；当输入信号从正变负时，结果信号应为 −1；其他情况下为 0。（假设 0 输入的符号为正。）例如，一段典型的输入信号及其对应的零交叉信号如下：

… 1 2 1.5 1 0.5 -0.1 -2 -3 -2 -0.5 0.2 3 4 …
… 0 0 0 0 0 -1 0 0 0 0 1 0 0 …

在 Alyssa 的系统中，传感器信号表示为流 sense-data，而流 zero-crossings 是对应的零交叉流。Alyssa 首先写了一个过程 sign-change-detector，它接受两个值作为参数，通过比较两值的符号来产生适当的 0、1 或 −1。然后她按如下方式构造零交叉流：

(define (make-zero-crossings
input-stream last-value)
(cons-stream
(sign-change-detector
(stream-car input-stream)
last-value)
(make-zero-crossings
(stream-cdr input-stream)
(stream-car input-stream))))

(define zero-crossings
(make-zero-crossings sense-data 0))

Alyssa 的上司 Eva Lu Ator 路过时建议说，这个程序大致等价于下面这个使用练习 3.50 中广义 stream-map 的版本：

(define zero-crossings
(stream-map sign-change-detector
sense-data
⟨expression⟩))

请通过补全 ⟨expression⟩ 来完成这个程序。