Page 75 - DSP Integrated Circuits

60 Chapter 3 Digital Signal Processing

algorithm for computing the DFT was invented by Cooley and Tukey in 1965
[14]. This new algorithm was referred to as the FFT (fast fourier transform).
FFTs are today widely used for digital signal processing.
Another important class of transforms is the DCT (discrete cosine transforms),
mainly used in data compression schemes for speech and image signals. The DCT
is also used for realization of filter banks that are used, for example, in frequency
division and time division multiplexers, so-called transmultiplexors. Fast algo-
rithms for computing the DCT have also been developed.
Both the FFT and the DCT can be implemented by using standard signal pro-
cessors or ASIC processors. Many different implementation schemes, based on the
direct mapping approach, have also been developed. The implementations range
from fully parallel, where one PE (processing element) is allocated to each basic
operation, to sequential computations using a single processor.
In this chapter, we will discuss two typical fixed-function subsystems that will
be used as case studies. The first case study involves the design of an FFT proces-
sor for computing the DFT. The second case study concerns a two-dimensional
DCT that is intended to be used in a system for transforming coding of video
images in real-time. Such coding systems will be used for HDTV (high-definition
television).
The third case study will be discussed in Chapter 4.

3.2 DIGITAL SIGNAL PROCESSING

Most developments in advanced signal processing for signals with bandwidths up
to 50 MHz or more have since the early 1990s been based on discrete-time or digital
techniques. Typical applications are in audio, video, radar, sonar, digital radio, com-
munication, control, and measurement systems. Relatively simple signal process-
ing functions, which traditionally have been implemented using analog continuous-
time systems, have also received competition from discrete-time implementations
using SC (switched capacitor) techniques which are compatible with most modern
CMOS technologies.

3.2.1 Sensitivity
The major reason behind the increasing use of discrete-time and digital signal pro-
cessing techniques is that problems caused by component tolerances as well as
drift and aging of components are circumvented.
For analog frequency selective filters, realizations having minimum circuit
element sensitivity have been developed. Thus, by using high-quality components
high-performance filters can be implemented. However, there is a practical limit
to the performance of analog components, e.g., the tolerances of resistors, capaci-
tors, and amplifiers can not be arbitrarily low. Filters meeting very stringent
requirements are therefore not possible to implement. At the other end of the
spectrum, cheap and simple filters can be implemented using low-tolerance com-
ponents. No such lower tolerance bound exists for digital signal processing tech-
niques. In fact, the tolerances can easily be adjusted to the requirements at hand.
However, it must be stressed that the ultimate performance of a composite sys-