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**Digital Signal Processing**

*February 20, 2018*

Fixed versus Floating Point

Digital Signal Processing can be divided into two categories, fixed point, and floating point. These refer to the format used to store and manipulate numbers within the devices. Fixed-point DSPs usually represent each number with a minimum of 16 bits, although a different length can be used. For instance, Motorola manufactures a family of fixed point DSPs that use 24 bits. There are four common ways that these 216 = 65536 possible bit patterns can represent a number. In unsigned integer, the stored number can take on any integer value from 0 to 65,535. Similarly, signed integer uses two's complement to make the range include negative numbers, from -32,768 to 32,767. With unsigned fraction notation, the 65,536 levels are spread uniformly between 0 and 1. Lastly, the signed fraction format allows negative numbers, equally spaced between -1 and 1.

In comparison, floating point DSPs typically use a minimum of 32 bits to store each value. This results in many more bit patterns than for fixed point, 232 = 4,294,967,296 to be exact. A key feature of floating point notation is that the represented numbers are not uniformly spaced.

All floating point DSPs can also handle fixed-point numbers, a necessity to implement counters, loops, and signals coming from the ADC and going to the DAC. However, this doesn't mean that fixed point math will be carried out as quickly as the floating point operations; it depends on the internal architecture. For instance, the SHARC DSPs are optimized for both floating point and fixed point operations and executes them with equal efficiency. For this reason, the SHARC devices are often referred to as "32-bit DSPs," rather than just "Floating Point."

The primary trade-offs between fixed and floating point DSPs.

The fixed point arithmetic is much faster than floating point in general purpose computers. However, with DSPs, the speed is about the same, a result of the hardware being highly optimized for math operations. The internal architecture of a floating point DSP is more complicated than for a fixed point device. All the registers and data buses must be 32 bits wide instead of only 16; the multiplier and ALU must be able to quickly perform floating point arithmetic, the instruction set must be larger (so that they can handle both floating and fixed point numbers), and so on. Floating point (32 bit) has better precision and a higher dynamic range than a fixed point (16 bit). In addition, floating point programs often have a shorter development cycle, since the programmer doesn't generally need to worry about issues such as overflow, underflow, and round-off error.

On the other hand, fixed-point DSPs have traditionally been cheaper than floating point devices. Nothing changes more rapidly than the price of electronics; anything you find in a book will be out-of-date before it is printed. Nevertheless, the cost is a key factor in understanding how DSPs are evolving, and we need to give you a general idea.

What can a 32-bit floating point system do, that a 16-bit fixed point can't?

The answer to this question is signal-to-noise ratio. Suppose we store a number in a 32-bit floating point format. As previously mentioned, the gap between this number and its adjacent neighbor is about one ten-millionth of the value of the number. To store the number, it must be round up or down by a maximum of one-half the gap size. In other words, each time we store a number in floating point notation, we add noise to the signal.

The same thing happens when a number is stored as a 16-bit fixed point value, except that the added noise is much worse. This is because the gaps between adjacent numbers are much larger. For instance, suppose we store the number 10,000 as a signed integer (running from -32,768 to 32,767). The gap between numbers is one ten-thousandth of the value of the number we are storing. If we want to store the number 1000, the gap between numbers is only one one-thousandth of the value.

Noise in signals is usually represented by its standard deviation. For here, the important fact is that the standard deviation of this quantization noise is about one-third of the gap size. This means that the signal-to-noise ratio for storing a floating point number is about 30 million to one, while for a fixed point number it is only about ten-thousand to one. In other words, floating point has roughly 30,000 times less quantization noise than a fixed point.

This brings up an important way that DSPs are different from traditional microprocessors. Suppose we implement an FIR filter in fixed point. To do this, we loop through each coefficient, multiply it by the appropriate sample from the input signal, and add the product to an accumulator. Here's the problem. In traditional microprocessors, this accumulator is just another 16 bit fixed point variable. To avoid overflow, we need to scale the values being added, and will correspondingly add quantization noise on each step. In the worst case, this quantization noise will simply add, greatly lowering the signal-to-noise ratio of the system. For instance, in a 500 coefficient FIR filter, the noise on each output sample may be 500 times the noise on each input sample. The signal-to-noise ratio of ten-thousand to one has dropped to a ghastly twenty to one.

DSPs handle this problem by using an extended precision accumulator. This is a special register that has 2-3 times as many bits as the other memory locations. For example, in a 16 bit DSP, it may have 32 to 40 bits, while in the SHARC DSPs it contains 80 bits for fixed point use. This extended range virtually eliminates round-off noise while the accumulation is in progress. The only round-off error suffered is when the accumulator is scaled and stored in the 16-bit memory. This strategy works very well, although it does limit how some algorithms must be carried out. In comparison, floating point has such low quantization noise that these techniques are usually not necessary.

In addition to having lower quantization noise, floating point systems are also easier to develop algorithms for. Most DSP techniques are based on repeated multiplications and additions. In fixed point, the possibility of an overflow or underflow needs to be considered after each operation. The programmer needs to continually understand the amplitude of the numbers, how the quantization errors are accumulating, and what scaling needs to take place. In comparison, these issues do not arise in floating point; the numbers take care of themselves (except in rare cases).

How do you choose which to use?

Here are some things to consider. First, look at how many bits are used in the ADC and DAC. In many applications, 12-14 bits per sample is the crossover for using fixed versus floating point. For instance, television and other video signals typically use 8 bit ADC and DAC, and the precision of fixed point is acceptable. In comparison, professional audio applications can sample with as high as 20 or 24 bits, and almost certainly need a floating point to capture the large dynamic range.

The next thing to look at is the complexity of the algorithm that will be run. If it is relatively simple, think fixed point; if it is more complicated, think floating point. For example, FIR filtering and other operations in the time domain only require a few dozen lines of code, making them suitable for fixed point. In contrast, frequency domain algorithms, such as spectral analysis and FFT convolution, are very detailed and can be much more difficult to program. While they can be written in fixed point, the development time will be greatly reduced if floating point is used.

How important is the cost of the product, and how important is the cost of the development?

When fixed point is chosen, the cost of the product will be reduced, but the development cost will probably be higher due to the more difficult algorithms. In a reverse manner, floating point will generally result in a quicker and cheaper development cycle, but a more expensive final product.

Fixed point is more popular in competitive consumer products where the cost of the electronics must be kept very low. A good example of this is cellular telephones. When you are in competition to sell millions of your product, a cost difference of only a few dollars can be the difference between success and failure. In comparison, floating point is more common when greater performance is needed and the cost is not important. For instance, suppose you are designing a medical imaging system, such a computed tomography scanner. Only a few hundred of the model will ever be sold, at a price of several hundred thousand dollars each. For this application, the cost of the DSP is insignificant, but the performance is critical. In spite of the larger number of fixed point DSPs being used, the floating point market is the fastest growing segment.

A fixed point usually use 32 bits and 16 bits, respectively, but not always. For instance, the SHARC family can represent numbers in 32-bit fixed point, a mode that is common in digital audio applications. This makes the 232 quantization levels spaced uniformly over a relatively small range, say, between -1 and 1. In comparison, floating point notation places the 232 quantization levels logarithmic-ally over a huge range, typically ±3.4×1038. This gives 32-bit fixed point better precision, that is, the quantization error on any one sample will be lower. However, the 32-bit floating point has a higher dynamic range, meaning there is a greater difference between the largest number and the smallest number that can be represented.

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