fixed point multiplication for normal multiplication

Question

I need to multiply X with a floating point number in floating point as i don't have floating point operations in my processor. I understand the method but don't know why that method exists?

Suppose we want to multiply 2*4.5 in decimal I do the below: 2 * 4.5 (100.1) So i multiply 2*1001 = 2*9 = 18 and then right shift by 1. so 18>>1 = 9

Is it that we represent 2 in fixed point and represent 4.5 in fixed point and as we multiply Q1.1 and Q1.1 format so we get Q2.2 format and we do right shifting causing Q1.1 format result.Is this right?

Answer 1

In decimal, your fixed-point example is actually:

2 * 4.5
2 * 45 (after multiplying by 10) = 90
90 / 10 = 9 (after dividing the 10 back out)

In binary, the same thing is being done, but just with powers of 2 instead of powers of 10 (as the factors / divisors). Fixed point operations occur in purely integral space after appropriate multiplications. And multiplying or dividing by a power of 2 is just a left shift or right shift respectively on the binary number (very fast for the CPU). In fixed-point the number of bits to the left (integer) and right (fractional) of the decimal point are fixed (predetermined), which means that some numbers cannot be represented on the scale without loss of precision.

Floating-point further extends the concept by allowing the number of bits assigned to the left and right of the decimal point to be flexible. In floating point, every number is represented as an integral "significand" (or mantissa) to a specified power (for example, a power of 2). This representation allows the same number of significant digits to be maintained over a greater dynamic range (for very small or very large magnitude numbers). For floating point, most of the bits will be assigned to the significant digits of the mantissa, and fewer of the bits assigned to the digits of the power. Floating-point calculations are more expensive (time-wise) than fixed-point, which is why fixed-point remains popular in microcontrollers and embedded systems.

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