Bitmasking in DSP Code

[martinvicanek]

Howdee,

the subject of using bitmasks in FS e.g. to implement logical structures comes up every now and then so I thought perhaps a tutorial might be helpful to have. I plan to do this in a number of installments so this is only the first part of a series. Please feel free to interrupt me and post questions.

It is always a good idea to consult the User Guide first, so let me start with that:

ConditionalStatements.png (41.4 KiB) Viewed 44188 times

Let's elaborate a bit. There is the & operator, a "bitwise and" for the arguments left and right of it. In the example given, one argument is a boolean expression (x >= 1) and the other is a float number 1.0. Both have a 32 bit representation in FS per SSE channel. The boolean expression is either true or false:

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true  = 11111111111111111111111111111111
false = 00000000000000000000000000000000

Float numbers are represented as bit patterns according to the IEEE 754 standard, which for the example happens to be

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1.0 = 00111111100000000000000000000000

"Bitwise and" means that each of the 32 binary digits is evaluated to 1 if both those digits are 1, and is zero otherwise. At the end of the day, all you need to remember is the following:

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number & true  = number
number & false = 0

By the way, the order of the arguments does not matter, so bool & number is the same as number & bool.

[Spogg]

This is going to be interesting and useful!

Thanks

Spogg

[RJHollins]

Time for some schoolin'. Thanks Martin

[martinvicanek]

So let's look at some working examples.
Our first example will be a (naive, non bandlimited) sawtooth oscillator with frequency f and the waveform going from -1 to +1.

C code would look something like this:

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float f = 0.1;
float saw = 0;

saw = saw + f;
if (saw > 1.0) saw = saw - 2.0;

This translates to the following FS code:

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float f = 0.1;
float saw = 0;

saw = saw + f;
saw = saw - 2&(saw > 1);

Easy, isn't it?

Next consider a simple index ramp. A C code would look something like this:

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integer i = 0;
integer n = 10;

i = i + 1;
if (i >= n) i = 0;

The FS code box offers only floating point declarations, so it would read

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float i = 0;
float n = 10;

i = i + 1;
i = i&(i < n);

Note that we have negated the comparison in FS in this case.

So how do we implement an if-then-else structure? For instance

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if (x > y) then z = a;
else z = b;

In FS you could write

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z = a&(x > y) + b&(x <= y);

This expression is correct, however it involves two comparisons. A more efficient implementation is

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z = b + (a - b)&(x > y);

which requires only one comparison. To see why this gives the same result assume (x > y) is true, then z = b + (a - b) = a. If, on the other hand, (x > y) is false then z = b + 0 = b.

Note: You could also write

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z = a + (b - a)&(x <= y);

[tulamide]

First of all: Thank you for taking the time to start at the very beginning!

Unfortunately, I already am overchallenged with the last two code examples. While it is easy to comprehend when it is already there, I just don't know how I would get from the two-condintional version to the one-conditional version on my own. I'm missing the steps involved to get from

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z = a&(x > y) + b&(x <= y);

to

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z = b + (a - b)&(x > y);

Because that doesn't come naturally to me. But exactly those are the things that make code more efficient.

[martinvicanek]

Look at it this way: In the instruction

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z = b + (a - b)&(x > y);

the first term b is the "default" value whereas the second term (a - b) represents a correction in the case that the condition (x > y) is true. In prose you could say:

If nothing else then set z equal to b.
However, in the event that (x > y) replace it by a.
Do so by adding a and subtracting b.

Of course you can reverse the roles of a and b if you prefer a to be the "default":

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z = a + (b - a)&(x <= y);

Did it help?

[adamszabo]

However to make things even more complicated, the first example with two comparisons is actually more efficient in assembly I believe. In assembly it could look like this:

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z = a&(x > y) + b&(x <= y);

movaps xmm0,x;       //we are moving 'x' to register xmm0
cmpps xmm0,y,2;      //we compare if x is <= y then we get the bitmask, either true or false
movaps xmm1,xmm0; //we store this bitmask to another register
andps xmm0,b;         //b value will be active if condition is met (and)
andnps xmm1,a;       //a is active if the inverse condition is met (and not)
andps xmm0,xmm1;   //then we choose the correct value

You can see we used only one comparison and also andnps so we didnt use any addition or subtraction which use more cpu than 'and' or 'and not'

[martinvicanek]

Adam, you are miles ahead of me!
Yes, if you do it in assembly you have more options, for instance you can use andnps. But watch out, the negation applies to the first argument! (You have that right in your code). The other thing is that it may not work with older FS versions, dunno when they added that opcode.

In (older) textbooks you read that bit operations are faster than floating point operations (and that floating point adds are faster than multiplies). I am not sure if this is still valid with modern CPUs, so I tested your ASM implemetation against my z = b + (a - b)&(x > y); ASM implementation - and they were tie!

Anyway, let's not frighten the audience with assembly nitpicking, there is still some DSP code material ahead.

[RJHollins]

and we [I] are easily frightened !

[martinvicanek]

The arguments of of the & operator need not be a number and a boolean, respectvely. For instance you might want to combine two booleans. An example is the following zero crossing detector:

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streamin in;    // current sample
float in1;      // previous sample
float zx;       // boolean, true if zero crossing just happened

zx  = (in > 0)&(in1 <= 0);
in1 = in;

Note that FS allows you to store booleans, even though the declared datatype is float.

In addition to the & operator, FS also has an "or" operator denoted by |. This can be useful to combine boolean expressions. Suppose you want to start off a timer triggered by an impulse at the input. you could use the above expression for zero crossing, however it is more efficient to work with the negated expression (in <= 0)|(in1 > 0):

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streamin in;    // input trigger pulse
float in1;      // previous input value
float time;     // elapsed time (in samples) since last trigger

time = time + 1;
time = time&((in <= 0)|(in1 > 0));   // reset timer on trigger
in1  = in;