Click here to go to the applet.
This java applet demonstrates
Fourier series,
which is a method of expressing an arbitrary periodic function as
a sum of cosine terms. In other words, Fourier series can be used to express a
function in terms of the frequencies (harmonics) it is composed of.
To select a function, you may press one of the following buttons:
Sine, Cosine, Triangle, Sawtooth, Square, and Noise. The function is
displayed in white, with the Fourier series approximation in red.
If you only see a red graph, that means the Fourier approximation is
nearly the same as the original function. (The red graph is drawn
on top of the white one.)
Below the function you will see a graph of the Fourier coefficients.
Each one represents a frequency, or harmonic.
There are two sets of terms; on top are the sine terms, and on the bottom
are the cosine terms. Low frequencies are on the left and
higher frequencies are on the right.
The "Number of Terms" slider will adjust the number of terms in the
expansion. The more terms there are, the
better the approximation. Try sliding the "Number of Terms" slider from left
to right slowly to see the Fourier terms added up one by one.
If you move the mouse over one of the harmonics, it will turn
yellow, and the corresponding Fourier term (frequency) will be
drawn on top of the function in yellow. So if you move the mouse
over all the harmonics, you can see each of the terms individually.
Also, if you click with the right mouse button on one of the harmonics,
the "Number of Terms" slider will be changed so that all higher-frequency
terms will be taken out of the series.
You can modify the function in one of two ways. You can
edit the function directly by clicking on it; in this case,
the Fourier coefficients will be regenerated when you are done.
Or, you can modify the Fourier coefficients, in which case the
function will be changed to match. If you want to create a function
from scratch, hit the "Clear" button.
The "Clip" button can be used to simulate clipping. It will
increase the amplitude of the function,
but will clip it if it goes out of range. Try doing this with a
sine function. Using this button generally results in more jagged
edges, which means more high-frequency terms will be involved in the
expansion. Some functions, like the square wave, aren't affected
much by clipping. Try playing the noise waveform and then hit "Clip"
a few times to see if it changes.
The "Resample" button can be used to demonstrate aliasing.
The function is replaced with samples at 32 points. This corresponds
to a sampling frequency of 7040 Hz (at the default playing frequency of
220 Hz). In almost all cases
this will cause
aliasing, where new, unwanted frequencies will be introduced. Press the
"Resample" button multiple times to choose a lower sampling rate.
For example
a frequency of 220 Hz, when resampled, will have aliases at 6820 Hz
(7040-220), 7260 Hz (7040+220), 13860 Hz (7040*2-220),
and so on.
When the "Quantize" button is pressed, the values in the function are rounded
off to the nearest multiple of 1/8. The resulting roundoff error causes
new frequencies to be introduced; these can be heard when the function
is played. This is called quantization distortion. Press "Quantize"
multiple times to use a coarser quantization.
The "Rectify" button will zero out the function wherever it is negative.
The "Full Rectify" button will take the absolute value of the function
at all points. Try this with the sine or sawtooth function.
The "Sound" checkbox, when checked, will play the function at 220 Hz.
This checkbox is only available if you have Java 2. The
frequency is adjustable using the "Playing Frequency" slider.
Try playing
various functions to see what they sound like. Also try isolating each
harmonic, or mixing two or three harmonics together. The "Number of
Terms" slider will affect how many harmonics are played.
(The sampling rate used for playing is 22.05khz, so frequencies higher
than 11025 Hz will be filtered out.)
There are two lines of letters at the bottom of the window which can be used
to easily “hear out” harmonics (when Sound is enabled). If you click on one
of the top lowercase 'm's, it capitalizes to "M", meaning "mute"; and that
harmonic will not be played. If you click on one of the bottom lowercase
's's, it capitalizes to "S", meaning "solo"; only that harmonic (along with
any other similarly soloed harmonics) will be played.
Clicking a letter again restores its previous state.
Numeric keys (not numpad digits) 1-0 can be used as short-cuts instead of clicking the 'm's to toggle the mute
attribute for each of the first ten harmonics.
Similarly, shifted presses of these same numeric keys toggle the corresponding solo attributes.
For instance, Shift+4, Shift+5, Shift+6 plays a major chord extracted from a complex tone.
The "Mag/Phase view" checkbox will allow you to view the Fourier terms
in phase-angle form (see
this
page for more on this representation).
The "Log View" checkbox (which only works when "Mag/Phase View" is checked)
will use a logarithmic scale to show the magnitudes of each term.
Other things to try:
- Draw a rapidly varying sine wave (as best you can), and check out
the Fourier coefficients to see if the frequency is picked up by the
Fourier decomposition. One of the magnitudes should be much higher
than the others.
- Start with a function, and remove Fourier terms one by one by
setting the coefficients to zero.
- Start with a sine wave, and then hit the "Clip" button repeatedly.
Watch how higher-frequency terms become more and more prominent.
- Hit "Clear" and then add two frequencies to the function by
clicking on the magnitudes. See how the frequencies interact.
If the frequencies are fairly close together, you should see beats;
the amplitude of the overall function will oscillate at a frequency
which is equal to the difference in the original two frequencies.
Click here to go to the applet.
java@falstad.com
A far higher place must be assigned to Judaism among the competitors for the allegiance of Europe. The cosmopolitan importance at one time assumed by this religion has been considerably obscured, owing to the subsequent devolution of its part to Christianity. It is, however, by no means impossible that, but for the diversion created by the Gospel, and the disastrous consequences of their revolt against Rome, the Jews might have won the world to a purified form of their own monotheism. A few significant circumstances are recorded showing how much influence they had acquired, even in Rome, before the first preaching of Christianity. The first of these is to be found in Cicero’s defence of Flaccus. The latter was accused of appropriating part of the annual contributions sent to the temple at Jerusalem; and, in dealing with this charge, Cicero speaks of the Jews, who were naturally prejudiced against his client, as a powerful faction the hostility of which he is anxious not to provoke.330 Some twenty years later, a great advance has been made. Not only must the material interests of the Jews be respected, but a certain conformity to their religious prescriptions is considered a mark of good breeding, In one of his most amusing satires, Horace tells us how, being anxious to shake off a bore, he appeals for help to his friend Aristius Fuscus, and reminds him of217 some private business which they had to discuss together. Fuscus sees his object, and being mischievously determined to defeat it, answers: ‘Yes, I remember perfectly, but we must wait for some better opportunity; this is the thirtieth Sabbath, do you wish to insult the circumcised Jews?’ ‘I have no scruples on that point,‘ replies the impatient poet. ‘But I have,’ rejoins Fuscus,—‘a little weak-minded, one of the many, you know—excuse me, another time.‘331 Nor were the Jews content with the countenance thus freely accorded them. The same poet elsewhere intimates that whenever they found themselves in a majority, they took advantage of their superior strength to make proselytes by force.’332 And they pursued the good work to such purpose that a couple of generations later we find Seneca bitterly complaining that the vanquished had given laws to the victors, and that the customs of this abominable race were established over the whole earth.333 Evidence to the same effect is given by Philo Judaeus and Josephus, who inform us that the Jewish laws and customs were admired, imitated, and obeyed over the whole earth.334 Such assertions might be suspected of exaggeration, were they not, to a certain extent, confirmed by the references already quoted, to which others of the same kind may be added from later writers showing that it was a common practice among the Romans to abstain from work on the Sabbath, and even to celebrate it by praying, fasting, and lighting lamps, to visit the synagogues, to study the law of Moses, and to pay the yearly contribution of two drachmas to the temple at Jerusalem.335 Jeff’s hand was quietly coming down. “What happened to you?” begged Sandy. “Something new has come up, sir. I was waiting there by my ship a good while back, and I heard another one cruising and spiraling, shooting the field, I guess, because he came in and set down. My crate, just the way you ordered, was down by the grove, not in plain sight in the middle of the course. But Jeff set his ship down, left the engine running, and went off. I stayed hid to see what would happen, but when he didn’t come back, I thought I’d better go and find you—and see if it meant anything to you.” She stood alone, with the sticky, wet knife in her hand, catching her breath, coming out of the madness. Then she stooped, and pushing the branches aside felt about for her pistol. It lay at the root of a tree, and[Pg 80] when she had picked it up and put it back in the holster, there occurred to her for the first time the thought that the shot in the dead stillness must have roused the camp. And now she was sincerely frightened. If she were found here, it would be more than disagreeable for Landor. They must not find her. She started at a swift, long-limbed run, making a wide detour, to avoid the sentries, bending low, and flying silently among the bushes and across the shadowy sands. The year 1756 opened with menaces to England of the most serious nature. The imbecility of the Ministry was beginning to tell in the neglect of its colonies and its defences. France threatened to invade us, and a navy of fifty thousand men was suddenly voted, and an army of thirty-four thousand two hundred and sixty-three of native troops; but as these were not ready, it was agreed to bring over eight thousand Hessians and Hanoverians. To pay for all this it was necessary to grant excessive supplies, and lay on new duties and taxes. In presenting the money bills in the month of May, Speaker Onslow could not avoid remarking that there were two circumstances which tended to create alarm—foreign subsidies and foreign troops introduced, and nothing but their confidence in his Majesty could allay their fears, or give them confidence that their burdens would be soon reduced. There was, in fact, no chance for any such reduction, for wars, troubles, and disgraces were gathering around from various quarters. The first reverse came from the Mediterranean. MUCH to their amazement, the boys waked up the next morning in Nashville, and found that they had passed through the "dark and bloody ground" of Kentucky absolutely without adventure. After drawing and dividing the rations and cartridges. Si gave the boys the necessary instruction about having their things ready so that they could get them in the dark the next morning, and ordered them to disregard the bonfires and mirth-making, and lie down to get all the sleep they could, in preparation for the hard work of the next day. Then, like the rest of the experienced men, who saw that the campaign was at length really on, and this would be the last opportunity for an indefinite while to write, he sat down to write short letters to his mother and to Annabel. "Bully for the Wild Wanderers of the Wabash," Shorty joined in. "They're the boss regiment in the army o' the Cumberland, and the Army o' the Cumberland's the boss army on earth. Hooray for US Co. Le's have a speech. Where's Monty Scruggs?" "Bring a light, do—I can't abide this dark." Albert suddenly began to look uneasy. After all he was not really drunk, only a little fuddled. He walked straight, and his roll was natural to him, while though he was exceedingly cheerful, and often burst into song, his words were not jumbled, and he generally seemed to have a fair idea of what he was saying. "But I heard what the doctor said to you." "A purty accident—wud them stacks no more dry than a ditch. 'Twas a clear case of 'bustion—fireman said so to me; as wicked and tedious a bit o' wark as ever I met in my life." Calverley stept from the shadow of the cliff, and beheld a meteor in the sky, brightening and expanding, as the clouds opened, until it assumed the appearance of a brilliant star, of astonishing magnitude, encircled by dazzling rays, which, in a singular manner, were all inclined in one direction, and pointing to that part of the horizon where lay the rival of England—France. The foreman's face assumed a deeper hue than usual: he looked fiercely at the galleyman, but there was a determination in the weather-beaten face that made him pause ere he spoke. "Galleyman," he at length said, "you knew the business before you came: if you be so fond of saving old witches' lives, why didn't you say so, that I might not now be in this dilemma?" "No, no, not the boy," replied Merritt, rather impatiently.