CH05 Fixed- Point versus Floating-Point

# CH05 Fixed- Point versus Floating-Point - CHAPTER 5...

This preview shows pages 1–3. Sign up to view the full content.

C H A P T E R 5 Fixed-Point versus Floating-Point From an arithmetic point of view, there are two ways a DSP system can be implemented in LabVIEW to match its hardware implementation on a processor. These include fixed-point and ﬂoating-point implementations. In this chapter, we discuss the issues related to these two hardware implementations. In a fixed-point processor, numbers are represented and manipulated in integer format. In a ﬂoating-point processor, in addition to integer arithmetic, ﬂoating-point arithmetic can be handled. This means that numbers are represented by the combination of a mantissa (or a fractional part) and an exponent part, and the processor possesses the necessary hardware for manipulating both of these parts. As a result, in general, ﬂoating-point processors are slower than fixed-point ones. In a fixed-point processor, one needs to be concerned with the dynamic range of num- bers, since a much narrower range of numbers can be represented in integer format as compared to ﬂoating-point format. For most applications, such a concern can be virtually ignored when using a ﬂoating-point processor. Consequently, fixed-point processors usually demand more coding effort than do their ﬂoating-point counterparts. 5.1 Q-format Number Representation The decimal value of an N -bit 2’s-complement number, B ¼ b N ± 1 b N ± 2 ::: b 1 b 0 ; b i 2f 0 ; 1 g , is given by D ð B Þ¼± b N ± 1 2 N ± 1 þ b N ± 2 2 N ± 2 þ ::: þ b 1 2 1 þ b 0 2 0 (5.1) The 2’s-complement representation allows a processor to perform integer addition and subtraction by using the same hardware. When the unsigned integer repre- sentation is used, the sign bit is treated as an extra bit. This way, only positive numbers can be represented. 123

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
There is a limitation of the dynamic range of the foregoing integer representation scheme. For example, in a 16-bit system, it is not possible to represent numbers larger than 2 15 ± 1 ¼ 32767 and smaller than ± 2 15 ¼ ± 32768. To cope with this limitation, numbers are often normalized between –1 and 1. In other words, they are represented as fractions. This normalization is achieved by the programmer moving the implied or imaginary binary point (note that there is no physical memory allocated to this point) as indicated in Figure 5-1 . This way, the fractional value is given by F ð B Þ ¼ ± b N ± 1 2 0 þ b N ± 2 2 ± 1 þ ::: þ b 1 2 ±ð N ± 2 Þ þ b 0 2 ±ð N ± 1 Þ (5.2) This representation scheme is referred to as the Q-format or fractional representa- tion. The programmer needs to keep track of the implied binary point when manipulating Q-format numbers. For instance, let us consider two Q15 format numbers. Each number consists of 1 sign bit plus 15 fractional bits. When these numbers are multiplied, a Q30 format number is generated (the product of two fractions is still a fraction), with bit 31 being the sign bit and bit 32 another sign bit (called an extended sign bit). Assuming a 16-bit wide memory, not enough bits are available to store all 32 bits, and only 16 bits can be stored. It makes sense to store
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern