Jun 152014
 

Everything that happens in the world can be described in some way. Our descriptions range from informal and causal to precise and scientific, yet ultimately they all share one underlying characteristic: they carry an abstract idea known as “information” about what is being described. In building complex systems, whether out of people or machines, information sharing is central for building cooperative solutions. However, in any system, the rate at which information can be shared is limited. For example, on Twitter, you’re limited to 140 characters per message. With 802.11g you’re limited to 54 Mbps in ideal conditions. In mobile devices, the constraints go even further: transmitting data on the network requires some of our limited bandwidth and some of our limited energy from the battery.

Obviously this means that we want to transmit our information as efficiently as possible, or, in other words, we want to transmit a representation of the information that consumes the smallest amount of resources, such that the recipient can convert this representation back into a useful or meaningful form without losing any of the information. Luckily, the problem has been studied pretty extensively over the past 60-70 years and the solution is well known.

First, it’s important to realize that compression only matters if we don’t know exactly what we’re sending or receiving beforehand. If I knew exactly what was going to be broadcast on the news, I wouldn’t need to watch it to find out what happened around the world today, so nothing would need to be transmitted in the first place. This means that in some sense, information is a property of things we don’t know or can’t predict fully, and it represents the portion that is unknown. In order to quantify it, we’re going to need some math.

Let’s say I want to tell you what color my car is, in a world where there are only four colors: red, blue, yellow, and green. I could send you the color as an English word with one byte per letter, which would require 3, 4, 5, or 6 bytes, or we could be cleverer about it. Using a pre-arranged scheme for all situations where colors need to be shared, we agree on the convention that the binary values 00, 01, 10, and 11 map to our four possible colors. Suddenly, I can use only two bits (0.25 bytes, far more efficient) to tell you what color my car is, a huge improvement. Generalizing, this suggests that for any set \chi of abstract symbols (colors, names, numbers, whatever), by assigning each a unique binary value, we can transmit a description of some value from the set using \log_2(|\chi|) bits on average, if we have a pre-shared mapping. As long as we use the mapping multiple times it amortizes the initial cost of sharing the mapping, so we’re going to ignore it from here out. It’s also worthwhile to keep this limit in mind as a max threshold for “reasonable;” we could easily create an encoding that is worse than this, which means that we’ve failed quite spectacularly at our job.

But, if there are additional constraints on which symbols appear, we should probably be able to do better. Consider the extreme situation where 95% of cars produced are red, 3% blue, and only 1% each for yellow and green. If I needed to transmit color descriptions for my factory’s production of 10,000 vehicles, using the earlier scheme I’d need exactly 20,000 bits to do so by stringing together all of the colors in a single sequence. But, given that by the law of large numbers, I can expect roughly 9,500 cars to be red, so what if I use a different code, where red is assigned the bit string 0, blue is assigned 10, yellow is assigned 110, and green 111? Even though the representation for two of the colors is a bit longer in this scheme, the total average encoding length for a lot of 10,000 cars decreases to 10,700 bits (1*9500 + 2*300 + 3*100 + 3*100), almost an improvement of 50%! This suggests that the probabilities for each symbol should impact the compression mapping, because if some symbols are more common than others, we can make them shorter in exchange for making less common symbols longer and expect the average length of a message made from many symbols to decrease.

So, with that in mind, the next logical question is, how well can we do by adapting our compression scheme to the probability distribution for our set of symbols? And how do we find the mapping that achieves this best amount of compression? Consider a sequence of n independent, identically distributed symbols taken from some source with known probability mass function p(X=x), with S total symbols for which the PMF is nonzero. If n_i is the number of times that the i th symbol in the alphabet appears in the sequence, then by the law of large numbers we know that for large n it converges almost surely to a specific value: \Pr(n_i=np_i)\xrightarrow{n\to \infty}1 .

In order to obtain an estimate of the best possible compression rate, we will use the threshold for reasonable compression identified earlier: it should, on average, take no more than approximately \log_2(|\chi|) bits to represent a value from a set \chi , so by finding the number of possible sequences, we can bound how many bits it would take to describe them. A further consequence of the law of large numbers is that because \Pr(n_i=np_i)\xrightarrow{n\to \infty}1 we also have \Pr(n_i\neq np_i)\xrightarrow{n\to \infty}0 . This means that we can expect the set of possible sequences to contain only the possible permutations of a sequence containing n_i realizations of each symbol. The probability of a specific sequence X^n=x_1 x_2 \ldots x_{n-1} x_n can be expanded using the independence of each position and simplified by grouping like symbols in the resulting product:

P(x^n)=\prod_{k=1}^{n}p(x_k)=\prod_{i=1}^{S} p_i^{n_i}=\prod_{i=1}^{S} p_i^{np_i}

We still need to find the size of the set \chi in order to find out how many bits we need. However, the probability we found above doesn’t depend on the specific permutation, so it is the same for every element of the set and thus the distribution of sequences within the set is uniform. For a uniform distribution over a set of size |\chi| , the probability of a specific element is \frac{1}{|\chi|} , so we can substitute the above probability for any element and expand in order to find out how many bits we need for a string of length n:

B(n)=-\log_2(\prod_{i=1}^Sp_i^{np_i})=-n\sum_{i=1}^Sp_i\log_2(p_i)

Frequently, we’re concerned with the number of bits required per symbol in the source sequence, so we divide B(n) by n to find H(X) , a quantity known as the entropy of the source X , which has PMF P(X=x_i)=p_i :

H(X) = -\sum_{i=1}^Sp_i\log_2(p_i)

The entropy, H(X) , is important because it establishes the lower bound on the number of bits that is required, on average, to accurately represent a symbol taken from the corresponding source X when encoding a large number of symbols. H(X) is non-negative, but it is not restricted to integers only; however, achieving less than one bit per symbol requires multiple neighboring symbols to be combined and encoded in groups, similarly to the method used above to obtain the expected bit rate. Unfortunately, that process cannot be used in practice for compression, because it requires enumerating an exponential number of strings (as a function of a variable tending towards infinity) in order to assign each sequence to a bit representation. Luckily, two very common, practical methods exist, Huffman Coding and Arithmetic Coding, that are guaranteed to achieve optimal performance.

For the car example mentioned earlier, the entropy works out to about 0.35 bits, which means there is significant room for improvement over the symbol-wise mapping I suggested, which only achieved a rate of 1.07 bits per symbol, but it would require grouping multiple car colors into a compound symbol, which quickly becomes tedious when working by hand. It is kind of amazing that using only ~3,500 bits, we could communicate the car colors that naively required 246,400 bits (=30,800 bytes) by encoding each letter of the English word with a single byte.

H(X) also has other applications, including gambling, investing, lossy compression, communications systems, and signal processing, where it is generally used to establish the bounds for best- or worst-case performance. If you’re interested in a more rigorous definition of entropy and a more formal derivation of the bounds on lossless compression, plus some applications, I’d recommend reading Claude Shannon’s original paper on the subject, which effectively created the field of information theory.

Jan 012014
 

For a client’s website, I needed to enumerate the 12 months preceding a given date to create links to archived content. The site uses a javascript templating engine to create HTML, offloading the process from the server, so generating the list of months on the client side in javascript seemed like a reasonable choice. For the past week, everything looked great, but suddenly today I noticed that it was repeating the current month.

The code itself is pretty simple. It’s written as a dustjs helper function that uses the Date.setMonth function to handle wrapping from January of one year to December of the previous year.

dust.helpers.month_list = function(chunk, ctx, bodies, params) {
	var count = dust.helpers.tap(params.count, chunk, ctx) | 0;
	var curDate = new Date();

	for (var i = 0; i < count; ++i) {
		var dateStr = (1900 + curDate.getYear()) + '-' + (curDate.getMonth()+1);
		chunk = chunk.render(bodies.block, ctx.push({date : dateStr}));
		curDate.setMonth(curDate.getMonth()-1);
	}

	return chunk;
}

Some quick printf debugging, adding console.log(curDate) at the start of the loop, shows this surprising result:

Tue Dec 31 2013 20:47:14 GMT-0800 (Pacific Standard Time)
Sun Dec 01 2013 20:47:14 GMT-0800 (Pacific Standard Time)
Fri Nov 01 2013 20:47:14 GMT-0700 (Pacific Daylight Time)
Tue Oct 01 2013 20:47:14 GMT-0700 (Pacific Daylight Time)

Apparently, on the 31st of the month, subtracting one month from the current date does not have the desired effect, in Chrome (on Windows 8.1). I ran the test again in IE 11 and observed the same behavior, as well as tried by manually setting the date to the 31st of October and subtracting a month, again seeing the same behavior. I'm not sure if that means this is somehow part of the specification, or if it's a bug caused by a library used underneath the hood in both browsers, but the end result is the same. My "clean" code to use setMonth(getMonth()-1) instead of writing out an if statement to detect the start of the year and wrap correctly now contains a cryptic loop that detects if the month didn't actually change and subtracts it again, all to deal with a bug that only happens once a month.

Jul 202013
 

There aren’t very many good resources for optimizing compute-bound JavaScript on the Internet, aside from this article from Google Developers discussing string building and touching very briefly on some of the problems with functions. For the most part, I’d bet this is because a lot of developers say “we’ll leave it up to the JIT compiler to do optimizations,” and then they write code the idiomatic way, assuming that this will translate into high performance, and the rest focus on non-compute usage, which is generally limited in performance by networked resources. However, as I’ve found, the v8 optimizer, as good as it is, still struggles with a number of situations that are easily handled in C. Obviously, I’m not the first to notice this, with projects like asm.js that exist to identify a subset of standard JavaScript that can achieve high performance by allowing for better optimization and especially by avoiding garbage collection.

However, I think that there are a number of things that can be done without resorting to asm.js, to increase performance. I’ve assembled four test cases/demonstrations to illustrate some problematic areas for the JIT optimizer and identify how I’ve hand-optimized them. All of the test cases will be discussed in terms of simple benchmark results, using Node.js’s high resolution timer to achieve a higher precision. One thing that is important to note is that sometimes the relative speed increase appears huge (for instance 10x faster), but really we are looking at a very simple, pared down test case. In this instance it is much more meaningful to consider the absolute speedup (5 microseconds vs. 10 microseconds), because when used in real code, the function bodies will generally have far more meat on them, but the same amount of overhead will be eliminated.

My attempts at optimizing a graph processing library that I have been working on identified function calls to be the biggest source of overhead, because the optimizer doesn’t appear to do much with them. I haven’t tried to look at tracing or disassembly, because I want to keep this simple for now. Since functions appear to be the problem, let’s start with a simple test: does v8 inline functions?

var iter = 10000;
var i, j, start, stop;

function inc(a) { return a+1; }

var x = 0;
start = process.hrtime();
for (i = 0; i < iter; ++i) {
	x = inc(x);
}
stop = process.hrtime(start);

time = (stop[0]*1e9 + stop[1])/1e6;
console.log('T1//with function: '+time+' ms');

var x = 0;
start = process.hrtime();
for (i= 0; i < iter; ++i) {
	x = x + 1;
}
stop = process.hrtime(start);

time = (stop[0]*1e9 + stop[1])/1e6;
console.log('T1//hand inlined: '+time+' ms');

For me, the version calling inc() generally takes about 0.467 milliseconds to complete, while the hand-inlined version takes only 0.0365 ms. The answer, then, is that either v8 doesn't inline, or if it does, it cannot match a hand-inlined version of the same code. I am not knowledgeable enough about the engine internals or about the specification requirements for JavaScript to speculate on why this happens, but for now, I am satisfied to notice the problem and try to work around it in performance-critical code.

However, this actually leads to a serious problem from a design perspective. You can either write well-partitioned code that is easy to read and follow, with good compartmentalization for function reuse, or you can stuff it all into a single function and expect a significant amount of call overhead to be eliminated. I think this could be (easily?) rolled into a tool like UglifyJS, to automatically inline functions, but at that point, we're introducing a build step for an interpreted language. For scripts that will be sent to the browser, this is perfectly acceptable, as we generally would want to pass these through UglifyJS (although you shouldn't use a separate build step for this anyway), but for server-side scripts, this is extremely tedious. Yes, you can automate it with grunt or another task runner, but it just reintroduces the exact type of process we are trying to eliminate by using JavaScript instead of C.

For now, I'll just say that I think that some amount of function inlining should be possible for the JIT optimizer to do, but there are a lot of instances (member functions, for instance, due to the completely dynamic way in which methods are called) where this is not possible. Therefore, let's move on, and examine the performance of JavaScript's iteration constructs.

One of the things that really attracted me to Node.js was Underscore.js, a functional programming library that provides the types of things I would normally find in Scheme. The best thing about these tools is that they make it simple to write clean, easy to maintain code, and they help enforce good design principles by encouraging programmers to write small functions for one task, and then to build larger processes out of those functions. It might be obvious in hindsight, but considering that function optimization is not very effective, using functions to do array iteration is probably slower than using a for loop to do the same thing. Let's look at two variations, one testing Array.reduce and one testing Array.forEach in an O(n^2) situation to see how the function overhead scales with nested loops.

// Test 2: Iteration methods vs. for loops
iter = 1000;
var elems = 100, subelems = 10, sum;
var xa = [];

for (i = 0; i < elems; ++i) {
	xa[i] = Math.random();
}

// 2a: array sum with reduce
start = process.hrtime();
sum = xa.reduce(function(memo, v) {
	return memo + v;
}, 0);
stop = process.hrtime(start);

time = (stop[0]*1e9 + stop[1])/1e6;
console.log('T2A//using reduce: '+time+' ms');

start = process.hrtime();
sum = 0;
for (i = 0; i < xa.length; ++i) {
	sum += xa[i];
}
stop = process.hrtime(start);

time = (stop[0]*1e9 + stop[1])/1e6;
console.log('T2A//for loop: '+time+' ms');
console.log('');

// On my machine, times fluctuate a lot, but 0.10-0.16 ms vs 0.0054-0.0078 ms
// so the for loop is faster again

// 2b: nested iteration, calling a member function for each element of an array
function t2b_class() {
	this.val = Math.random();
}
var t2b_sum = 0;
t2b_class.prototype.work = function() {
	t2b_sum += this.val;
};

for (i = 0; i < elems; ++i) {
	xa[i] = [];
	for (j = 0; j < elems; ++j) {
		xa[i][j] = new t2b_class();
	}
}

start = process.hrtime();
xa.forEach(function(v) {
	v.forEach(function(v2) {
		v2.work();
	});
});
stop = process.hrtime(start);

time = (stop[0]*1e9 + stop[1])/1e6;
console.log('T2B//nested iterators: '+time+' ms');

function inner_iter(v2) {
	v2.work();
}
start = process.hrtime();
xa.forEach(function(v) {
	v.forEach(inner_iter);
});
stop = process.hrtime(start);
time = (stop[0]*1e9 + stop[1])/1e6;
console.log('T2B//nested iterators, avoids redeclaration: '+time+' ms');

start = process.hrtime();
for (i = 0; i < xa.length; ++i) {
	for (j = 0; j < xa[i].length; ++j) {
		xa[i][j].work();
	}
}
stop = process.hrtime(start);

time = (stop[0]*1e9 + stop[1])/1e6;
console.log('T2B//for loop: '+time+' ms');

For Test 2A, I find that using Array.reduce takes between 0.10 and 0.16 milliseconds, while using the for loop to implement a reduce operation "the long way" takes between 0.0045 and 0.0075 ms. These really aren't huge differences, but in principle they shouldn't exist at all, as, from a semantic point of view, the code does exactly the same thing in both implementations. For test 2B, however, there is a much more significant difference. For the nested forEaches, I've tested two possible implementations: the "obvious" approach, which uses an anonymous inner function for the second loop, and the "observant" approach which realizes that this function gets redeclared every time the loop iterates, so the declaration is moved outside of the outer loop. My results are that the obvious approach is the slowest, coming in at around 1.08-1.32 ms, then the observant approach, coming in around 0.87-0.95ms, and then the explicit, nested for loop blowing both out of the water with a time of 0.22-0.35 ms.

The results for 2A are underwhelming, but 2B illustrates a much more significant problem with the use of functional techniques. It's really easy to write code in a functional style that ends up being really slow, because v8 does not effectively optimize it. For me, this is a disaster. Functional-style code is faster to write, easier to maintain, and generally cleaner than its imperative counterpart. Plus, implementing a map, reduce, or even just a for-each loop using the for construct and explicit iteration requires repeating a lot of glue code that is handled internally. Even C++11 has language features to minimize this kind of repetition, now. As with function inlining, I maintain that for both 2A and 2B, these types of transformations can easily be done by a tool like UglifyJS at the source level, but really, I feel that the JIT optimizer should be capable of doing them as well.

Now, let's look at another type of optimization important for performance when using recursive functions: tail call optimization. I've written the canonical factorial function in three styles: first, a basic recursive implementation that works its way back up the stack to compute its result; second, a version that uses tail-calls and could therefore be optimized without any source transformation; and third, a complete iterative version, to determine whether v8 performs maximally effective tail optimization.

// Test 3: Tail calls vs. stack walking vs. iterative method
function fact_tail(n, accumulator) {
	if (n < 2)
		return accumulator;
	return fact_tail(n-1, accumulator*n);
}
function fact_stack(n) {
	if (n < 2)
		return 1;
	return n * fact_stack(n-1);
}
function fact_iter(n) {
	var accumulator = n;
	for (n = n-1; n > 1; --n) {
		accumulator = accumulator*n;
	}
	return accumulator;
}

iter = 1000;
var n = 30, result;

start = process.hrtime();
for (i = 0; i < iter; ++i) {
	result = fact_tail(n-1, n);
}
stop = process.hrtime(start);
time = (stop[0]*1e9 + stop[1])/1e6;
console.log('T3//Tail Factorial: '+time+' ms');

start = process.hrtime();
for (i = 0; i < iter; ++i) {
	result = fact_stack(n);
}
stop = process.hrtime(start);
time = (stop[0]*1e9 + stop[1])/1e6;
console.log('T3//Stack Factorial: '+time+' ms');

start = process.hrtime();
for (i = 0; i < iter; ++i) {
	result = fact_iter(n);
}
stop = process.hrtime(start);
time = (stop[0]*1e9 + stop[1])/1e6;
console.log('T3//Iterative Factorial: '+time+' ms');

The results are approximately what you'd expect. The tail call is not fully optimized, but it is faster than the version using the stack. I haven't done enough research to determine whether the speed boost is due to (some) actual optimization being performed or if it's just a by-product of how temporary variables are used implicitly in the local scope. The numbers for me come in at 0.515 ms for the basic stack version, 0.277 ms for the tail-call version, and 0.181 ms for the iterative version.

Again we see that a source transformation from "good, easy-to-follow code" to "awkward, but still makes sense if you stop to think about it code" results in improved performance. This is the same type of trade-off that we faced in the 90s when C compilers did not optimize effectively. And, again, I would posit that this is the type of operation that should be done by the JIT optimizer. If only I had the necessary knowledge to actually try to modify v8 myself--I've looked at the code, and for all my complaints about shortcomings, it is both incredibly impressive and far too complicated for me to "pick up" over the weekend and change).

Finally, I have one last optimization case. This one, however, is not something that can be done by a JIT optimizer, because the result would no longer meet the language specification. For those of you familiar with C++, there are two basic types of methods in a class: "normal" methods, which have a fixed address known at compile time, and "virtual" methods, which are resolved at run time based on the virtual function table stored within the object on which the method is being called (I almost said "dynamically determined by the type" but that would be inaccurate, as virtual methods do not require RTTI). In JavaScript all class methods are implicitly of a virtual type. When you call a function, that field on the object must be looked up every time, because it can be changed at any time. This is an incredibly useful feature of the language, because of the flexibility it provides: polymorphism is "easy," objects can be monkey-patched to modify behavior, and some of the more subtle aspects of C++ go away (for instance, do you know off the top of your head which method is called if you dereference a base class-type pointer to an instance of a derived class and then call a virtual method?). But this power comes at a cost. When a method isn't used in this manner, we still pay the price of doing lookups, and because the optimizer cannot know that we will not replace a method definition, it can't substitute in the address of the specific function we want.

There is actually a simple, clever way to obtain static references to functions. When the method is called through the object as a property lookup, the prototype chain must be traversed to find the first matching function and then it is used. If we skip that step and use the specific function, on the specific prototype that we want, it should be slightly faster. We still pay the price of looking up a property on the prototype object, though, so we can even take this a step further and save a variable that points precisely to the function we want to call. Because of JavaScript's ability to call any function with any "this" context, we can take the static function pointer and call it as though it were a member function without going through the dynamic lookup phase. Let's look at a test to compare this.

// Test 4: Function table overhead
function TestClass(a, b, c) {
	this.a = a;
	this.b = b;
	this.c = c;
}
TestClass.prototype.hypot = function() {
	return Math.sqrt(this.a*this.a + this.b*this.b + this.c*this.c);
}

iter = 10000;
var fn = TestClass.prototype.hypot;
xa = new TestClass(Math.random(), Math.random(), Math.random());

start = process.hrtime();
for (i = 0; i < iter; ++i) {
	result = xa.hypot();
}
stop = process.hrtime(start);
time = (stop[0]*1e9 + stop[1])/1e6;
console.log('T4//Object method lookup: '+time+' ms');

start = process.hrtime();
for (i = 0; i < iter; ++i) {
	result = TestClass.prototype.hypot.call(xa);
}
stop = process.hrtime(start);
time = (stop[0]*1e9 + stop[1])/1e6;
console.log('T4//Prototype method lookup: '+time+' ms');

start = process.hrtime();
for (i = 0; i < iter; ++i) {
	result = fn.call(xa);
}
stop = process.hrtime(start);
time = (stop[0]*1e9 + stop[1])/1e6;
console.log('T4//Pre-saved method pointer: '+time+' ms');

For me, the results are that full dynamic lookup takes 0.194 ms, retrieving the method through the prototype takes 0.172 ms, and using a pre-saved pointer to the function takes only 0.135 ms. These absolute differences really aren't that significant, but this can be applied to every single non-polymorphic function call in your project, which means this might be the test case where relative speedup (30% faster with static function pointers) matters more than absolute. If you still think I'm crazy for even mentioning this, consider the extremely common problem of converting an arguments variable into a proper array by calling Array.prototype.slice on it. Many people will save a reference to Array.prototype.slice at the top of their module and use it throughout their code; luckily it's both shorter to type AND faster, which is great for libraries that provide a lot of functions accepting variable numbers of arguments.

Member function lookup optimization requires far more work than the other three, and it cannot be applied transparently by a JIT optimizer or by UglifyJS as the language specification prevents it from having enough information to correctly do so. It also makes your code much, much more annoying to read. I could suggest writing an AltJS language that allows you to differentiate between the two types of member functions, so that when it compiles to JS it spits out static references where possible, but I'm pretty sure @horse_js would make fun of me for it (rightly so). But, if you're in a situation where performance for computation is just that important, I'd recommend at least considering static function references.

As for my graph library, by carefully rewriting code matching the four above cases, I was able to get the execution time for a single pass of the evaluator down from 25.5 usec to 7.4 usec, which increases the number of users that my application can support for a given amount of hardware by 3.3x, immediately translating into lower operating costs. I wish I could share the code (maybe one day, but for now it's under the "secret project" category), but hopefully an unsubstantiated anecdote like this will help convince you that these optimizations still benefit real code that performs real tasks, and not just overly-simple test cases. There is often an important tradeoff to consider between readability, maintainability, and performance, so I wouldn't recommend always writing code that looks like this, but if you're working on a compute-bound application (mostly, JavaScript-based games, which we will see many of as WebGL and HTML5 are improved), maybe think about writing your code in three steps: first, do it the easy/fast/right way, producing good, clean code. Then, write a very good set of test cases. Finally, optimize your code to death, and use your test cases to verify that you haven't broken anything.

The complete benchmark file used to generate my numbers is available as a gist in case you don't feel like cobbling together the snippets given above.

Jul 062013
 

The Wiener filter is well known as the optimal solution to the problem of estimating a random process when it is corrupted by another additive process, using only a linear combination of values of the measured process. Mathematically, this means that the Wiener filter constructs an estimator of some original signal x(t) given z(t)=x(t)+n(t) with the property that \|\hat{x}(t)-x(t)\| is minimized among all such linear estimators, assuming only that both x(t) and n(t) are stationary and have known statistics (mean, variance, power spectral density, etc.). When more information about the structure of x(t) is known, different estimators may be easier to implement (such as a Kalman filter for signals with a recursive structure).

Such a filter is very powerful—it is optimal, after all—when the necessary statistics are available and the input signals meet the requirements, but in practice, signals of interest are never stationary (rarely even wide sense stationary, although it is a useful approximation), and their statistics change frequently. Rather than going through the derivation of the filter, which is relatively straightforward and available on Wikipedia (linked above), I’d like to talk about how to adapt it to situations that do not meet the filter’s criteria and still obtain high quality results, and then provide a demonstration on one such signal.

The first problem to deal with is the assumption that a signal is stationary. True to form for engineering, the solution is to look at only a brief portion of the signal and approximate it as stationary. This has the unfortunate consequence of preventing us from defining the filter once and reusing it; instead, as the measured signal is sliced into approximately stationary segments, we must estimate the relevant statistics and construct an appropriate filter for each segment. If we do the filtering in the frequency domain, then for segments of length N we are able to do the entire operation with two length N FFTs (one forward and one reverse) and O(N) arithmetic operations (mostly multiplication and division). This is comparable to other frequency domain filters and much faster than the O(N^2) number of operations required for a time domain filter.

This approach creates a tradeoff. Because the signal is not stationary, we want to use short time slices to minimize changes. However, the filter operates by adjusting the amplitude of each bin after a transformation to the frequency domain. Therefore, we want as many bins as possible to afford the filter high resolution. Adjusting the sampling rate does not change the frequency resolution for a given amount of time, because the total time duration of any given buffer is f_{s}N. So, for fixed time duration, the length of the buffer will scale inversely with the sampling rate, and the bin spacing in an FFT will remain constant. The tradeoff, then, exists between how long each time slice will be and how much change in signal parameters we wish to tolerate. A longer time slice weakens the stationary approximation, but it also produces better frequency resolution. Both of these affect the quality of the resulting filtered signal.

The second problem is the assumption that the statistics are known beforehand. If we’re trying to do general signal identification, or simply “de-noising” of arbitrary incoming data (say, for sample, cleaning up voice recorded from a cell phone’s microphone in windy area, or reducing the effects of thermal noise in a data acquisition circuit), then we don’t know what the signal will look like beforehand. The solution here is a little bit more subtle. The normal formulation of the Wiener filter, in the Laplace domain, is

G(s)= \frac{S_{z,x}(s)}{S_{z}(s)}
\hat{X}(s)=G(s) Z(s)

In this case we assume that the cross-power spectral density, S_{z,x}(s), between the measured process z(t) and the true process x(t) is known, and we assume that the power spectral density, S_{z}(s), of the measured process z(t) is known. In practice, we will estimate S_{z}(s) from measured data, but as the statistics of x(t) are unknown, we don’t know what S_{z,x}(s) is (and can’t measure it directly). But, we do know the statistics of the noise. And, by (reasonable) assumption, the noise and the signal of interest are independent. Therefore, we can calculate several related spectra and make some substitutions into the definition of the original filter.

S_z(s)=S_x(s)+S_n(s)
S_{z,x}(s)=S_x(s)

If we substitute these into the filter definition to eliminate S_x(s), then we are able to construct and approximation of the filter based on the (known) noise PSD and an estimate of the signal PSD (if the signal PSD were known, it’d be exact, but as our PSD estimate contains errors, the filter definition will also contain errors).

G(s)=\frac{S_z(s)-S_n(s)}{S_z(s)}

You may ask: if we don’t know the signal PSD, how can we know the noise PSD? Realistically, we can’t. But, because the noise is stationary, we can construct an experiment to measure it once and then use it later. Simply identify a time when it is known that there is no signal present (i.e. ask the user to be silent for a few seconds), measure the noise, and store it as the noise PSD for future use. Adaptive methods can be used to further refine this approach (but are a topic for another day). It is also worth noting that the noise does not need to be Gaussian, nor does it have any other restrictions on its PSD. It only needs to be stationary, additive, and independent of the signal being estimated. You can exploit this to remove other types of interference as well.

One last thing before the demo. Using the PSD to construct the filter like this is subject to a number of caveats. The first is that the variance of each bin in a single PSD estimate is not zero. This is an important result whose consequences merit more detailed study, but the short of it is that the variance of each bin is essentially the same as the variance of each sample from which the PSD was constructed. A remedy for this is to use a more sophisticated method for estimating the PSD by combining multiple more-or-less independent estimates, generally using a windowing function. This reduces the variance and therefore improves the quality of the resulting filter. This, however, has consequences related to the trade-off between time slice length and stationary approximation. Because you must average PSDs computed from (some) different samples in order to reduce the variance, you are effectively using a longer time slice.

Based on the assigned final project in ECE 4110 at Cornell University, which was to use a Wiener filter to de-noise a recording of Einstein explaining the mass-energy equivalence with added Gaussian white noise of unknown power, I’ve put together a short clip comparing the measured (corrupted) signal, the result after filtering with a single un-windowed PSD estimate to construct the filter, and the result after filtering using two PSD estimates with 50% overlap (and thus an effective length of 1.5x the no-overlap condition) combined with a Hann window to construct the filter. There is a clear improvement in noise rejection using the overlapping PSD estimates, but some of the short vocal transitions are also much more subdued, illustrating the tradeoff very well.

Be warned, the first segment (unfiltered) is quite loud as the noise adds a lot of output power.

Here is the complete MATLAB code used to implement the non-overlapping filter

 
% Assumes einsteindistort.wav has been loaded with
[d,r] = wavread('EinsteinDistort.wav');

% Anything that can divide the total number of samples evenly
sampleSize = 512;

% Delete old variables
% clf;
clear input;
clear inputSpectrum;
clear inputPSD;
clear noisePSD;
clear sampleNoise;
clear output;
clear outputSpectrum;
clear weinerCoefficients;

% These regions indicate where I have decided there is a large amount of
% silence, so we can extract the noise parameters here.
noiseRegions = [1 10000;
                81000 94000;
                149000 160000;
                240000 257500;
                347500 360000;
                485000 499000;
                632000 645000;
                835000 855000;
                917500 937500;
                1010000 1025000;
                1150000 116500];

% Now iterate over the noise regions and create noise start offsets for
% each one to extract all the possible noise PSDs
noiseStarts = zeros(length(noiseRegions(1,:)), 1);
z = 1;
for k = 1:length(noiseRegions(:,1))
    for t = noiseRegions(k,1):sampleSize:noiseRegions(k,2)-sampleSize
        noiseStarts(z) = t;
        z = z + 1;
    end
end

% In an effort to improve the PSD estimate of the noise, average the FFT of
% silent noisy sections in multiple parts of the recording.
noisePSD = zeros(sampleSize, 1);
for n = 1:length(noiseStarts)
    sampleNoise = d(noiseStarts(n):noiseStarts(n)+sampleSize-1);
    noisePSD = noisePSD + (2/length(noiseStarts)) * abs(fft(sampleNoise)).^2 / sampleSize;
end

% Force the PSD to be flat like white noise, for comparison
% noisePSD = ones(size(noisePSD))*mean(noisePSD);

% Now, break the signal into segments and try to denoise it with a
% noncausal weiner filter.
output = zeros(1, length(d));
for k = 1:length(d)/sampleSize
    input = d(1+sampleSize*(k-1):sampleSize*k);
    inputSpectrum = fft(input);
    inputPSD = abs(inputSpectrum).^2/length(input);
    weinerCoefficients = (inputPSD - noisePSD) ./ inputPSD;
    weinerCoefficients(weinerCoefficients < 0) = 0;
    outputSpectrum = inputSpectrum .* weinerCoefficients;

    % Sometimes for small outputs ifft includes an imaginary value
    output(1+sampleSize*(k-1):sampleSize*k) = real(ifft(outputSpectrum, 'symmetric'));
end

% Renormalize and write to a file
output = output/max(abs(output));
wavwrite(output, r, 'clean.wav');

To convert this implementation to use 50% overlapping filters, replace the filtering loop (below "Now, break the signal into segments…") with this snippet:

output = zeros(1, length(d));
windowFunc = hann(sampleSize);
k=1;
while sampleSize*(k-1)/2 + sampleSize < length(d)
    input = d(1+sampleSize*(k-1)/2:sampleSize*(k-1)/2 + sampleSize);
    inputSpectrum = fft(input .* windowFunc);
    inputPSD = abs(inputSpectrum).^2/length(input);
    weinerCoefficients = (inputPSD - noisePSD) ./ inputPSD;
    weinerCoefficients(weinerCoefficients < 0) = 0;
    outputSpectrum = inputSpectrum .* weinerCoefficients;

    % Sometimes for small outputs ifft includes an imaginary value
    output(1+sampleSize*(k-1)/2:sampleSize*(k-1)/2 + sampleSize) = output(1+sampleSize*(k-1)/2:sampleSize*(k-1)/2 + sampleSize) + ifft(outputSpectrum, 'symmetric')';
    k = k +1;
end

The corrupted source file used for the project can be downloaded here for educational use.

This can be adapted to work with pretty much any signal simply by modifying the noiseRegions matrix, which is used to denote the limits of "no signal" areas to use for constructing a noise estimate.

Jun 122013
 

I’ve been a huge fan of LISP and related programming languages (scheme, racket, etc.) for a very long time. I’ve been evangelizing the utility of learning how the language works and understanding functional concepts among my friends for almost as long, and recently a friend of mine finally caved and read McCarthy’s original paper. Once he realized how important it was, he decided to learn while solving the L-99, a list of 99 problems for LISP (based on Haskell’s list of 99 problems), except we use Scheme/Racket for our implementations, because of the excellent Dr. Racket environment on both Windows and Linux.

The purpose of the L-99 was to provide a series of challenges that would force someone to learn about the language, its syntax and structure, standard paradigms, etc., which something I think it accomplishes very well. But I think it also provides a great backdrop for discussing algorithms regardless of the language they are written in. L-99 therefore provides two opportunities: the first is learning how to do something in LISP, and the second is learning how to do it “the best way”—the one with the lowest time and/or memory complexities. An example I’d like to mention is the list reversal problem, #5, because it is easy to understand and has two easy solutions with very different characteristics. My friend’s solution (in Racket), which works correctly, is given below:

(define (list-reverse x)
  (cond
    [(null? (cdr x)) x]
    [else (append (list-reverse (cdr x)) (list (car x)))]))

As I mentioned above, his solution works correctly (aside from a little bit of parameter validation that is missing), but a close reading reveals that its time complexity is O(n^2). This is because the append function is O(n), and it is called once per item in the list to reverse. In another language, with another library (say, C++ with STL containers), appending to a list is a constant time operation, which is likely why he didn’t think much of using append here. That said, an opaque list structure can be made (in LISP) that offers constant time append operations, but this isn’t how a chain of cons cells works. For comparison, here is my solution to the problem, which uses a helper function to reverse the list in O(n) time:

(define (list-reverse-linear x)
  (define (lr-inner xa xr)
    (cond
      [(empty? (cdr xa)) (cons (car xa) xr)]
      [else (lr-inner (cdr xa) (cons (car xa) xr))]))
  (cond
    [(not (pair? x)) x]
    [(empty? (cdr x)) x]
    [else (lr-inner (cdr x) (cons (car x) null))]))

I suppose it is obvious to most people that faster asymptotic complexity is always better, but just for the sake of argument, let’s look at the time required for reversing a list with 10,000 elements in it:

; Input
(define (gen-list n)
  (cond
    [(= n 0) null]
    [else (cons n (gen-list (- n 1)))]))
(define tl (gen-list 10000))
(time-apply list-reverse (cons tl null))
(time-apply list-reverse-linear (cons tl null))

; Result is, in milliseconds (cpu real gc)
(3432 3427 2293)
(0 1 0)

A stark contrast—based on a measurement with time-apply, it takes nearly 3.5 seconds to reverse the list using the O(n^2) algorithm, but less than one millisecond to do so with the O(n) algorithm. A big part of this, of course, also comes from the garbage collection time, which accounts for nearly 2/3 of the run time in the slower algorithm, due to how many intermediate lists are generated and discarded, while the linear algorithm does not allocate any unused items. Another important difference is that my solution is tail-recursive, which enables significant optimizations (ones that Dr. Racket is capable of doing, too), compared to my friend’s solution.

I think that the L-99 has a wonderful secondary purpose. Not only can it be used to introduce someone to the standard idioms, tools, and techniques of functional programming, but it can also be used to introduce and discuss analysis of algorithms, as the problems can generally be solved in multiple ways, with different time and space complexity requirements. It even provides a platform to discuss different implementation strategies (such tail-recursion vs. not) and experiment with their effects on performance, all with problems and solutions that are very simple to state and understand. This is a great aspect of LISP as a tool for learning in general—it frees you as a programmer from worrying about necessary but conceptually irrelevant implementation details, to focus on understanding what you’re doing, why you’re doing it, and how you’re doing it, on an algorithmic level. Mastering the relevant skills translates readily into using other languages, where it helps separate the problems—algorithms, memory management, program structure, inheritance organization, the list goes on—out in your mind, allowing you to solve each individually.

Normally I’d write another 3+ pages, at minimum, trying to beat this to death, but I think this is a pretty self-defending position. Instead, I’d like to ask you a few questions to think about and maybe take with you as you do or learn about programming. Have you ever looked at the L-99? Can you solve all the problems, even if you don’t take the time to write out complete solutions? Are your solutions optimal in their asymptotic complexities? I’d also love to hear about any particularly clever solutions people have created.