Need help for searching "error upper bound" in my circuit design problem.

Hi:

I am an VLSI IC circuit design engineer. In my work, I encounter a math problem which I need help.

In my circuit, I have a variable T ( a real number) which can be expressed as:

T = FREQ * M

FREQ is a real number FREQ = I + r, I is the integer part, r is the fractional part, M is an integer.

However, in my circuit, I can't use factional number for FREQ. I can only use integers for this FREQ. The result is that I will have some error in T, as shown below.

If fractional numbers can be used for FREQ, then a given variable T can be achieved by:

T = FREQ * M = (I + r) * M

However, in real circuit, only integers can be used for FREQ, so T has to be approximated by T'

T' = FREQ' * M = I * M

the error

(T - T')/T = r/(I + r) <= 0.5/(I + 0.5)  since r <= 0.5

If FREQ' is in the range of I1 <= FREQ <= I2, then this error upper bound can be obtained as:

(T - T')/T <= 0.5/(I + 0.5) <= 0.5/(I1 + 0.5)


Now, I have a way of improving my circuit by allowing SOME fractional numbers. For example, for integer M, if it has factors, such as f1, f2, ...., fn. Then all the fractional numbers made of 1/f1, 1/f2, ... 1/fn can be used in my circuit.

For example, if M = 60, then its factors are: 2 ,3, 4, 5, 6, 10, 15, 20, 30. In this case: all the fractional number of 1/2, 1/3, 1/4, 1/5, 1/6, 1/10, 1/15, 1/20, 1/30 can be used in my circuit.

When I chose different M in the range of [M1 .. M2], I have to also adjust FREQ in the range of I1 <= FREQ <= I2 so that I can find a pair of FREQ and M that match the given T with least amount of error. This pair can be found easily by writing a searching program. However, I need a math prove of the upper bound.


Now the question:

FREQs is in the range of I1 <= I <= I2, M is in the range of M1 <= M <= M2, all the fractional numbers by the factors of M can be used in FREQ = I + r, then for a given required variable T, what is the error upper bound?


Thanks in advance!

Liming XIU
.

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