DETECTOR

In review

This page runs a BASIC program that detects the frequency of a periodic function. The program is rewritten in Python and runs right here in your browser. Edit the code and press Run to redraw the graph; Reset brings back the original program.

The function is \(f(t) = 3\sin(5t)\). For each frequency ω between 0 and 3, the program computes the average value of \(f(t)\sin(2\pi\omega t)\) over \(0 \le t \le 10\),

\[ \frac{1}{b - a} \int_a^b f(t)\, \sin(2\pi \omega t)\, dt, \]

and graphs that average against ω. When \(\sin(2\pi\omega t)\) rises and falls in step with \(f(t)\), the product is mostly positive and the average is large; at other frequencies the product keeps changing sign and the average stays near 0. The peak should sit at \(2\pi\omega = 5\), that is \(\omega = 5/(2\pi) \approx 0.796\), with height about 1.5, half of f’s amplitude.

There are two loops. The outer one, FOR j = 1 TO omegasteps, moves ω across from 0 to 3 in 1024 steps. The inner one, FOR k = 1 TO numberofsteps, finds the integral with a Riemann sum of 128 pieces. It starts at t = a + deltat / 2, the middle of the first piece, so this is a midpoint sum. Each pass of the outer loop draws a segment from the previous point (oldomega, oldaccum) to the new one.

Try changing fnf to 2 * math.sin(3 * t) + math.sin(8 * t) and look for two peaks, or shorten the time interval b to see the peak spread out.

QBasic lets a FUNCTION be written at the end of the program, after the code that uses it. Python needs a function to be defined before it is called, so in the Python the def comes first, and return gives back the value that BASIC sets with fnf = .... SIN becomes math.sin. The program gets \(2\pi\) as 8 * ATN(1), because the arctangent of 1 is \(\pi/4\) and QBasic has no built-in π; in Python that is 8 * math.atan(1), which is the same as 2 * math.pi.

Notes from the textbook

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Original BASIC program

 1  DEFDBL A-Z
 2  SCREEN 12
 3  WINDOW (0, -1)-(3, 2)
 4  startomega = 0
 5  endomega = 3
 6  omegasteps = 2 ^ 10
 7  deltaomega = (endomega - startomega) / omegasteps
 8  twopi = 8 * ATN(1)
 9  a = 0
10  b = 10
11  numberofsteps = 2 ^ 7
12  deltat = (b - a) / numberofsteps
13  omega = startomega
14  oldomega = omega
15  oldaccum = 0
16  FOR j = 1 TO omegasteps
17          t = a + deltat / 2
18          accum = 0
19          FOR k = 1 TO numberofsteps
20                  deltaS = (fnf(t) * SIN(twopi * omega * t) * deltat) / (b - a)
21                  accum = accum + deltaS
22                  t = t + deltat
23          NEXT k
24          omega = omega + deltaomega
25          LINE (oldomega, oldaccum)-(omega, accum)
26          oldomega = omega
27          oldaccum = accum
28  NEXT j
29  FUNCTION fnf (t)
30          fnf = 3 * SIN(5 * t)
31  END FUNCTION

Graph

Each line(...) draws one straight segment, like BASIC's LINE, and window(...) sets the visible range, like WINDOW. Hover over the graph to read coordinates.

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Python code

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