imation

;

is All you need

Arun Prakash A

Instructor

BS, IITM

Appro

\pi
import math 
PI = math.pi
=3.1415926

(Computer) Engineer

\pi
=3.14

Physicist

\pi
= \pi

Mathematician

Hilbert's name may

appear more often in the subsequent

talks!

No surprise!

Reasons for doing approximation

1.Storage and transmission

2.Computational efficiency

(In this talk)

Why do we care about storage in this modern era?

Most of the volume of internet data is occupied by

If data weights \(1 g\), then our earth would have turned into a black hole!

How I want you to see them..

f(t)
f(x,y)
f(x,y,t)

They are all  functions!

Running example

f(t)

Let's start with a simple function

f(t)
f(t) = ?
f(t) = \sin(4 \times 10^3 \pi t)
t (ms)

Let's hear it!

Well, the audio signals in the real world are continuous 

How do we store it in a digital device?

Discretize/sampling

 16000 samples/s

16 KB/s

56 MB for an hour of audio

280 hours of audio in 16GB Pendrive!

Wait a second, 

\(k=4 \times 10^3\)

f(t) = \sin(4 \times 10^3 \pi t)

Generator: \(A sin(k\pi t)\)

( could be An algorithm inside a computer)

\(A,K\) needs to be stored, it takes

8 Bytes  (32-bit processor)

Important Insight

If we have a well defined functional representation for the given waveform, we can store only their parameters such as \(A,k\) in the previous example

What about this function?

1
-1
T
f(t) = \begin{cases} 1 &\text{if } 0 < t <\frac{T}{2} \\ -1 &\text{if } \frac{T}{2} < t < T \end{cases}

Constraints: (for some reasons)

  • We can generate only sin waves
  • As many as we wish
  • Allowed to sum them up 

Generator: \(A sin(k\pi t)\)

( could be An algorithm inside a computer)

Let's Approximate

f(t)
f(t)
= a_1\sin(k\pi t)

Is it a good approximation?

No

Memory : 8 B

k
a_1

Let's Approximate

f(t)
f(t)
=a_1\sin(k\pi t)
a_3\sin(3k\pi t)

Let's Approximate

f(t)
f(t)
=a_1\sin(k\pi t)+a_3\sin(3k\pi t)
a_3\sin(3k\pi t)
+
+

Do not bother about where is \(a_2\) and why not \(2k\) instead of \(3k\)

Let's Approximate

f(t)
f(t)
=a_1\sin(k\pi t)+a_3\sin(3k\pi t)

Is it a good approximation?

No, but better than previous..

Memory : 16 B

k
a_1
2k
2k
a_3
Author: Steve phelps

Let's see what happens if we "Sum" more terms

f(t)
=a_1\sin(k\pi t)+a_3\sin(3k\pi t)+a_5\sin(5k \pi t)+ \cdots+a_{49}\sin(11k\pi t)+ \cdots

Memory : 200 B

f(t) = \begin{cases} 1 &\text{if } 0 < t <\frac{T}{2} \\ -1 &\text{if } \frac{T}{2} < t < T \end{cases}

Why not just store the definition of the  function itself?

Well, good question!.

True, as long as We know what the form of function is

But the real world wave form is more complex than these simple functions

Any unwillingness to learn mathematics today can greatly restrict your possibilities tomorrow

Richard Hamming