Electro Tech is an online community (with over 170,000 members) who enjoy talking about and building electronic circuits, projects and gadgets. To participate you need to register. Registration is free. Click here to register now.
Welcome to our site! Electro Tech is an online community (with over 170,000 members) who enjoy talking about and building electronic circuits, projects and gadgets. To participate you need to register. Registration is free. Click here to register now.
Could you please help me with **broken link removed**? It would be kind of you. I'm sorry if the answers are quite obvious for some of the queries. I'm really pressed for time and nothing is making any sense! Thank you.
Q1: The two methods (of course) do give different answers. It's hard to say which is preferred, but sometimes a real system only allows one method. For example, the backward calculation is likely to be the only one possible in real time if you are actively sampling on the fly. The forward value has not been sampled yet because it exists in the future. The only way to do the forward difference in real time systems is to delay the values, and then do the calculations. However, this introduces a one sample delay into the output, which is often not the best method. In such cases the backward difference is preferred. Image processing does not have this limitation if the difference is needed between pixels, which is a spatial derivative. Here you have more freedom, but perhaps here the trapezoidal rule is preferred then. Basically, it all depends ...
Q2 and Q3 are related and this subject directly ties in to the Fundamental Theorem of Calculus, which is the core of basic calculus and the foundation of the powerful Generalized Stoke's Theorem. Remember in calculus how the definite integral is calculated by evaluating the antiderivative at the endpoints of the boundary.
For Q2, basically, the end-points are the only points that don't cancel. It's strange that you ask this question here when you can actually see the terms cancel out, but you don't ask the same question when using the Fundamental Theorem of Calculus (or maybe you did ask). If the lower limit was not at minus infinity, you would have seen the lower endpoint also.
For Q3, the forward difference works too in principle, but write it out if you want to see the exact form.
Q1: I wouldn't say all systems output zero when power is off, but I think that's true of many systems. I also don't like what they said here. It is unnecessary to bring a power switch into the discussion. Systems are almost always models of something real. The system model makes assumptions about the real system in order to simplify the analysis, and usually the assumption is that power is on, and the system model may need to change drastically if the power is off.
Q2: Expression 2 is a summation of terms that are each always positive. Expression 1 is the sum of terms each with the same magnitude respectively compared to those in expression 2, but some terms can be positive and some terms can be negative, and then the final value of the sum is made positive with the absolute value. Hence, 2 is always greater if there are opposite signs in expression 1, and 2 is equal to 1 if all terms are the same sign in 1.
Could you please tell me of any simple system which doesn't output zero when power switch is off? That sounds weird to have a system which can have nonzero output even when it's not being fed in any power. Thanks.
Why does it sound weird that a system does not necessarily need power to provide a nonzero output?
First of all, there are passive circuits that don't need any power. Passive circuits can have energy storage devices and the stored energy can continue to provide power even when the inputs are switched to zero. This is not quite the same thing as turning off power, but it shows that a sudden shut off of the input can still allow an output. Now, take any active circuit that requires power and let it feed into a passive circuit with energy storage devices. When power goes off, that active circuit may drive the passive circuit with zero input, but the energy storage devices may then discharge and continue to provide an output while the power is off.
The simplest case is a circuit with a capacitor in parallel with the output. Eventually, we expect the energy to dissipate, but that does not happen instantly.
I would still like to discuss FIR and IIR for further clarification.
Q1: I believe a memoryless system produces FIR response. For example, a simple resistive circuit is a memoryless system therefore its impulse response is going to be FIR. But I believe in case of a simple resistor circuit FIR will die to zero almost instantly. What other system or circuit can you think of which is a FIR system but still its response take some time, like this one, to reach zero? Just curious!
Q2: A system with memory always produces IIR response and this response could ultimately die out to zero or it can become infinite. For example, a circuit with a capacitor in parallel with the output will start outputting zero as time t tends to infinity. But I can't think of an IIR system or circuit whose output never reaches a stage where it is zero or nearly zero. Could you please help me with it? Thanks.
Without delving too deeply into existing resources either on-line or from texts, I'll give my quick answers. You can obviously consult Google and your DSP books for answers that are strictly correct.
For analog systems, storage devices (caps and inductors) act like memory and create an IIR response typically. If you dont' have storage elements (i.e. resistors only), then you will get the FIR response. The tricky case here is delay lines. This is memory and storage that gives FIR (unless you put it in a feedback loop). Really, everything is IIR because you can never make a perfect delay line or a resistor without stray inductance and capacitance.
For digital systems, digital memory is not enough to give IIR responses, rather you need memory and feedback because feedforward memory eventually runs out. Unless you somehow implement a feedforward memory with an infinite number of delays. But, really you can't do that.
Q1: A lack of memory implies FIR for analog and digital. I think only digital filters will be FIR, when that system has memory, and then only when there is no feedback. So, you gave the example yourself when you pointed to "this one" in your question.
Q2: Generally, systems that do this are unstable. So, take any unstable system, and you will find it has an impulse response that blows up.
I understand that such a system whose IIR doesn't die out as time t tends to infinity is unstable. But I can't think of any such unstable system circuit. I think this text (check the paragraph just below the equation 4.16) gives a good practical example about a speaker and microphone system.
Though I have many other follow-on queries to make, I should wait for a proper time. Moreover, at this stage I believe I'm mixing continuous time systems and discrete time systems too much. Thanks.
It's easy to make unstable systems. There is an old joke that when you try to make an amplifier, you end up with an oscillator, and when you try to make an oscillator, you end up with an amplifier.
The lesson of this joke is that many systems, that you know about, are stable in theory (amps are stable in the linear sense and oscillators are stable in the nonlinear sense), but when implementing them, these real systems have parasitic effects that make the system different than you think. So, start building real circuits and you will find these unstable circuits and wish you never asked the question! (joking, hahaha )
OK, what about real circuits that are unstable without parasitics? ...
A typical opamp based amplifier circuit driving a capacitive load with too much capacitance will be unstable.
Or, take any first order system, add a delay to the output and then use proportional feedback from the output to the input. As you increase the gain of the proportional feedback, it will eventually go unstable.
Similarly, proportional feedback around a second order system will go unstable if the gain is too high.
Often, proportional-integral feedback is used in the above cases to try and improve control and have good stability margin. Still, if the P-gain and I-gain are not set correctly, instability is easily obtained.
For analog systems, storage devices (caps and inductors) act like memory and create an IIR response typically. If you dont' have storage elements (i.e. resistors only), then you will get the FIR response. The tricky case here is delay lines. This is memory and storage that gives FIR (unless you put it in a feedback loop). Really, everything is IIR because you can never make a perfect delay line or a resistor without stray inductance and capacitance.
Just wanted to clear a point. I think where you say that memory and storage gives FIR (in pink), you are referring to digital systems because before that you say (I think there you are referring to analog systems) that absence of storage elements gives FIR. Please let me know.
By the way, was there a special reason for using the words "memory and storage" together. I'm asking this because I take memory and storage a same thing. Please correct me if I'm wrong. Thanks a lot.
Yes, memory and storage are similar. In some contexts one word makes better sense than the other. This can apply to both digital and analog systems. I wasn't trying to be too precise in my wording. As I said above, you can consult books to get the perfect definitions and wordings. I learned them once, but I never found knowing this stuff too strictly to be terribly useful. So, I just gave a quick off-the-cuff perspective as I see it, without trying to relearn everything formally.
In reply to Q3 here (post #4), you said, "Yes". It means the system is unstable when a>1. First, is it a>1 or a>=0? How come it is an unstable system when a>1? In what sense it is unstable, BIBO or general stability criteria which is applicable to only LTI systems? In the attachment Reference #1 is helpful.
I even doubt the stability of moving average system which is thought to be a stable one because if the summation index, M, is too large, and if at the same time b_k and x[n] are also very large, then the sum might tend to be very large.
The case where a=1 is marginally unstable. I think both BIBO and LTI criteria apply here. It is unstable because the output diverges and is not bounded. Also, the pole is located outside the unit circle in the complex z-plane, if you use that test.
This site uses cookies to help personalise content, tailor your experience and to keep you logged in if you register.
By continuing to use this site, you are consenting to our use of cookies.
)
