ugbell@sunybcs.uucp (William Bell) (02/09/89)
I am taking a course at the University called:
Signal Processing Algorithms (ECE 416).
I have a strong mathematical background, and understand
the mathematics behind The Fourier and Z-Transforms. For example
I know that any signal is composed of an Amplitude and a
phase, and given any digital signal the signal can be changed
by using filters or Amplifiers. (I fully understand what Amplifiers
do - eg. Amplify the Power, Voltage or Current).
What I am confused about is: What is the purpose behind filter
design? I know they can be used in Radar. More specifically,
I know that FIR (finite impulse response) has the following
transfer function: H(z) = -1 -L
b + b z + ...+b z
0 1 L
Now, 1) What does it do? What are the benefits
over IIR? When is it used?
2) What is "impulse response"? I know it is h,
but is it just frequency??
3) For example they say that the impulse
response can be gotten by taking fourier,
and in FIR filters is an infinitely long
sequence, but what does this all mean?
4) I seen many diagrams saying "FIR Impulse
response for low-pass filter with..." and
Diagram shows Amplitude and Sample number,
but how can FIR be don to low pass "filter"?
2 filters?? hmmm...
Thanks.grnberg@mit-caf.MIT.EDU (David R. Greenberg) (02/12/89)
In article <4104@cs.Buffalo.EDU> ugbell@sunybcs.UUCP (William Bell) writes: > > I am taking a course at the University called: > Signal Processing Algorithms (ECE 416). > What I am confused about is: What is the purpose behind filter >design? >I know that FIR (finite impulse response) has the following >transfer function: H(z) = -1 -L > b + b z + ...+b z > 0 1 L > > Now, 1) What does it do? What are the benefits > over IIR? When is it used? > 2) What is "impulse response"? I know it is h, > but is it just frequency?? > 3) For example they say that the impulse > response can be gotten by taking fourier, > and in FIR filters is an infinitely long > sequence, but what does this all mean? > 4) I seen many diagrams saying "FIR Impulse > response for low-pass filter with..." and > Diagram shows Amplitude and Sample number, > but how can FIR be don to low pass "filter"? > 2 filters?? hmmm... > Thanks. I am usually a browser on this newsgroup and do not post. However, I felt guilty not responding to this question. I think, first of all, that you may not have as solid an idea of signals, transforms, and linear systems as you feel you do. Also, since I have time to answer these questions only briefly, I suggest that you do refer to a textbook such as _Digital Signal Processing_ by Jackson, or the signals and systems book by W.M. Siebert for further information. Although the algebra is often more difficult, most of the concepts of signals and linear systems are understood more easily with discrete time (DT) signals than with continuous time (CT) signals: A DT signal is just a sequence of numbers, x[n], where n is an integer. Usually, n ranges from negative to positive infinity. So, x[n] is really just a function that is defined only for an integer argument. As an example: x[n] = a^n, n>=0 0 , n<0 is a DT signal. Often, the term "digital" is used to describe this kind of signal. Strictly speaking, this term is not correct. A DT signal should be called digital only when x[n] is quantized and can take on only a discrete set of values e.g. the integers 0-255. Thus, both the domain and range of a digital signal are discrete. In real applications, we use a DT signal to represent information. For example, we may sample the voltage generated by a microphone at discrete instances of time. For example, x[0] could equal the voltage at 0 seconds, x[1] the voltage at 1 millisecond, and so forth. Because signals are used so often to represent quantities that evolve in time, we often refer to the sequence x[n] as the "time domain" representation of signal x. However, there are others ways to express the same information contained in signal x. One such way is called the z-transform, X(z). X(z) is a different function from x[n]. For one thing, z is a complex number, and X(z) itself is complex as well. Also, X(z) may be not be defined over the entire complex plane. In general, it is defined over the region contained between two concentric circles drawn in the complex plane (an annular region centered about 0). However, we can go back and forth between x[n] and X(z) with the appropriate mathematical formula - both functions contain the same information and should thought of as two different ways of expressing signal x. If we look at X(z) for z on the unit circle in the complex plane (assuming it is defined there), we obtain the frequency content of our signal. Now, it does us little good to be able to represent information in a DT signal if we cannot process this information. This is the purpose of filter design - to take a given signal and to process it in some way so as to extract information from it or to modify it in some desired way. As an example, you mention RADAR in your posting. Assume that we transmit a but of radio energy and pick up the "returned" signal with an antenna. Our DT signal would then represent the magnitude of the returned signal at discrete instances of time over some small period of time. Since this returned signal consists of noise as well as possible reflections from a target, we may wish to process the signal to determine if our signal does, in fact, contain reflections from actual targets. As another example, our signal may represent an audio signal. We may want to attenuate or amplify certain frequencies in this signal. We modify our original signal by putting it through a system. A DT system takes one signal, namely our original signal, processes it, and produces a second signal as its output. In your class, you will be learning about linear, time invariant (LTI) systems. We can think of such systems as modifying the frequency content of our original signal. This is why they are often called filters. We can represent LTI systems by their "system function", H(z). If original signal has z-transform X(z), than the output of our system will have z-transform Y(z)=X(z)H(z). We can represent the action of a filter in the time domain as well. In the time domain, output y[n] is just the convolution of our input x[n] and a signal h[n] called the impulse response of the system. The impulse reponse of a system (filter) is just the output of the system when the input is an impulse, i.e. the signal: delta[n] = 0, n not 0 1, n=0 Since any function can be written as a superposition of weighted, shifted impulses, the output of an LTI system can be written as a superposition of weighted, shifted impulse responses. The system function H(z) is just the z-transform of the impulse response. Now, suppose we apply an impulse as the input to a system. That is, we apply all zeros to the input until n=0, at which time we apply a single input of 1 followed by more zeros forever. Our input consists, therefore, of only one non-zero point. If our system is causal, the output of the system will be zero all the way until the point where we apply our single non-zero input. At this time, the system will begin to output non-zero values. If these non-zero values eventually die down and stop, our system is said to have a finite impulse response (FIR). If, on the other hand, our signal puts our non-zero values forever (e.g. an exponentially decaying sequence of values), our system has an infinite impulse response (IIR). Both FIR and IIR systems have uses. IIR filters can usually be of lower order than FIR filters that do the same job, and thus are easier to realize. However, IIR filters generally employ feedback which can lead to stability problems if they are not properly designed. IIR filters have another disadvantage over IIR filters. An FIR filter can be designed to have linear phase. That is, an FIR filter can be designed so that it delays all input frequencies by the same amount. Thus, the output, although delayed from the input, is not distorted by the spreading out of all of its frequency components. Compare this with a narrow pulse of white light. If the various colors of the light all travel at the same speed, the pulse will arrive intact at its destination some time later. If the colors each travel at different speeds, however, the pulse will be broadened at its destination, with the faster colors arriving first and the slower colors trailing behind. IIR filters cannot be designed to have linear phase. I have to give up my terminal to my officemate at the moment, so I can't go on. I'll try to keep up with the responses to this posting. -David