Digital Signal Processing Reference
In-Depth Information
CHAPTER
4
Transform and Filtering
Principles
4.1
OVERVIEW
Having become acquainted in the last chapter with basic signal acquisition (ADC) and reconstruction
techniques (DAC) and the all-important concepts of Nyquist rate and normalized frequency, we are now
in a position to investigate the powerful principle of
Correlation
. Discrete frequency transforms, which
can be used to determine the frequency content or response of a discrete signal, and time domain digital
filters (i.e., the FIR and IIR), which can preferentially select or reject certain frequencies in a signal,
function according to two principles of correlation-namely, respectively, the single-valued correlation of
two equally-sized, overlappingly aligned waveforms, and the
Correlation Sequence
. Another principle,
that of orthogonality, also plays an important role in frequency transforms, and this too will be explored
to show why frequency transforms such as the DFT require that two correlations (or a single complex
correlation) be performed for each frequency being tested.
We begin our discussion with an elementary concept of correlation, the correlation of two samples,
and quickly move, step-by-step, to the correlation of a signal of unknown frequency content with two
equally-sized orthogonal sinusoids (i.e., cosine-sine pairs) and, in short order, the real DFT. We then
briefly illustrate the use of the property of orthogonality in signal transmission, a very interesting topic
which illustrates the power of mathematics to allow intermixing and subsequent decoding of intelligence
signals. From there we investigate the correlation sequence, performing correlation via convolution, and
matched filtering. We then informally examine the frequency selective properties of the correlation se-
quence, and learn how to construct basic (although inefficient) filters. We learn the principle of sinusoidal
fidelity and then determination of time delays between sequences using correlation, an application of
correlation that is often used in echo canceller training and the like. For the final portion of the chap-
ter, we investigate simple one- and two-pole IIRs with respect to stability and frequency response, and
demonstrate how to generate IIRs having real-only filter coefficients by using poles in complex conjugate
pairs.
By the end of this chapter, the reader will be prepared to undertake the study of discrete frequency
transforms as found in Volume II in this series, which includes a general discussion of various transforms
and detailed chapters on the Discrete Time Fourier Transform (DTFT), the Discrete Fourier Transform
(DFT), and the
z
-Transform.
4.2
SOF TWARE FOR USE WITH THIS TOPIC
The software files needed for use with this topic (consisting of m-code (.m) files, VI files (.vi), and related
support files) are available for download from the following website:
http://www.morganclaypool.com/page/isen
