Expand All Collapse All


Activation function
Which Math library functions may I use?
Use Math.exp() and Math.abs(). You may also use the constants Math.POSITIVE_INFINITY, Math.NEGATIVE_INFINITY, and Math.NaN.
How do I detect whether x is NaN?
Use Double.isNaN(x). Do not write code like (x == Double.NaN), as that expression evaluates to false for all values of x, including NaN.
My softsign() function return NaN when x is positive infinity (or negative infinity) instead of 1 (or –1). Why is this?
You are dividing infinity by infinity, which leads to NaN (not a number). Include special cases for when x equals either Double.POSITIVE_INFINITY or Double.NEGATIVE_INFINITY,
My tanh() functions return NaN when x is larger than 750 (or smaller than –750) instead of 1 (or –1). Why is this?
The term \(e^x\) will be larger than the largest floating-point number. As a result, you end up dividing infinity by infinity, which leads to NaN. You may use the fact that if \(x \ge 20\), then tanh(x) should return 1.0; if \(x \le -20\), then tanh(x) should return -1.0.
Divisors
May I use recursion to implement gcd()?
That’s a fine idea, but please use a loop this week. We’ll learn about recursion next week.
How do I compute gcd() and lcm() when one (or both) of the arguments are negative?
It follows from the definitions that \(gcd(a, b) = gcd(|a|, |b|)\) and \(lcm(a, b) = lcm(|a|, |b|)\).
Why does Euclid’s algorithm correctly compute the greatest common divisor of two integers?
It is based on the observation that any common divisor d of the integers a and b must also divide the differences \(a - b, a - 2b, a - 3b, \ldots\), which includes \(a \; \% \; b\). Here’s a full explanation.
What should lcm() return if the least common multiple of the two arguments is larger than the largest int value?
We won’t test your program on such inputs, so you don’t need to worry about it here. In an industrial-strength implementation, you would probably throw an ArithmeticException to indicate the arithmetic overflow; we’ll learn about exceptions later in this course. An alternative would be to change the API to return the result as a long (instead of an int).
How many iterations does Euclid’s algorithm take to compute the greatest common divisor of a and b?
As you would expect, the number of iterations depends on a and b. It turns out that the smallest pair of integers that causes Euclid’s algorithm to take n iterations are the consecutive Fibonacci numbers \(F_{n+2}\) and \(F_{n+1}\). This implies that the number of steps is at most \(5 \log_{10} b\), i.e., at most 5 times the number of decimal digits in the smaller number.
Why do we use the conventions that gcd(0, 0) = 0 and lcm(0, k) = 0?
These are consistent with the more technical definitions of gcd and lcm as the meet (greatest lower bound) and join (least upper bound) in the lattice of divisibility.
Are there faster algorithms for computing Euler’s totient function?
Yes. Computing the totient function of an integer n is computationally equivalent to factoring n. Specifically, if the distinct prime factors of n are \(p_1, p_2, \ldots, p_r\), then \(\phi(n) = n \, (1 - \frac{1}{p_1}) \, (1 - \frac{1}{p_2}) \, (1 - \frac{1}{p_r})\). So, while computing the totient function is believed to be a computationally intractable problem for large n, it can be done substantially faster than brute-force approach adopted in this assignment.
Audio collage
What should each method do if one (or more) of the input samples is not between –1 and +1?
That’s ok. Just handle it as usual. While you should not play samples whose absolute value is greater than 1, it’s fine to manipulate such values along the way.
Can amplify() or mix() produce samples whose absolute value is larger than 1, even if all of the input samples are between –1 and +1?
Yes. While you should not play samples whose absolute value is greater than 1, it’s fine to produce them as intermediate results.
What does the mathematical notation ⌊x⌋ mean?
It’s the floor function—the largest integer less than or equal to \(x\). You can compute it in Java by either casting to an integer (resulting in an int) or calling Math.floor(x) (resulting in a double).
How can I convert an audio file into an appropriate format for use with StdAudio?
Convert it to a WAV file. Be sure to use 16-bit audio, monaural, and a sampling rate of 44,100 Hz. You may use an online conversion utility, such as Online-Convert.
What is a WAV file?
It is popular file format for storing raw and uncompressed audio data.
How do I read in the samples from a WAV file?
Use code like 
double[] samples = StdAudio.read("cow.wav");
Note that the autograder prohibts accessing files unless they are in the same directory as your .java file; do not use operating system specific file paths.
The changeSpeed() function changes not only the speed of the sound, but also the pitch? Is there a way to change the speed without affecting the pitch?

Speeding up the sound using resampling raises the pitch and leads to the Chipmunk effect. More sophisticated time stretching techniques are preferred in practice (such as when watching Coursera videos at 1.5× or 2× speed) because they change the speed but not the pitch.
Which other audio effects might I want to implement?

You could also synthesize your own sounds by creating a sine wave, square wave, triangle wave, or sawtooth wave of a given amplitude, frequency, and duration.