Several students were absent today, so Baraniuk polled the class to see how many people might have swine flu or have attended the Career Fair. Seeing the Career Fair as more threatening than H1N1 Flu, Baraniuk advocated the pursuit of graduate studies and suggested that undergraduate degrees are becoming commoditized and that BSEEs end up doing C Programming or facing job insecurity.
We don’t have to take Differential Equations at Rice for EE any more, but we use them all the time. We can relate the output and some of its derivatives to the input and some of its derivatives. This is an LTI system because there are no squared terms.
We can use an impulse response in order to determine the characteristics of a channel, which is a strict concept. For example, if we use a hand clap as an approximation of an impulse response and use the received signal of the listeners in the room (n ears), there are n channels with n possibly nonunique channels. Other approximations include racegun shots and other explosions. It’s important to have a known impulse approximation, however, because we can’t measure it in a pure form for each application. Also, we might be able to break down different types of known impulse responses in order to isolate desired systems. For example, a room has its own impulse response, but we just want to measure how a soundproofing panel absorbs sound. If we know the impulse response of the room without the panel and with the panel, can find the impulse response of the panel on its own. This is important if we want to determine isolated performance specs or if we want to model how the panel will perform in other configurations.
An LTI system is linear and time invariant. Linearity means that a scaled input leads to a scaled output. A time shifted input leads to an output time shifted in the same amount.
With a typical system, we take an input, f(t), send it through an LTI system with an impulse response h, and measure a signal y(t).
Convolution carries the following properties:
It’s Commutative, which means that it doesn’t matter in which order you list the operands. It’s Associative, meaning that if you need to convolve three items, Convolve 2 of them, then convolve the remainder. It’s also distributive, meaning that if you’re going to convolve with two added signals, you could instead convolve them separately and then add. Finally, it’s Time Invariant. When you convolve a signal with the impulse, you get the signal back (helpful for deconvolution). A way of visualizing convolution is by flipping one signal and slowly moving it through another signal, taking the area of the overlapping sections as the value of the output at that point. Accordingly, keep track of your t value.
Two things
1: Convolution, LTI, properties of convolution, limits of RLC components,
2: No Questions
On Thursday, we began to develop an intuitive feel for convolution. In short, to do convolution, you flip one signal around time=0, shift the flipped signal, then take the area of the overlapping region as the value of the convolution at that point. It’s important to break a convolution problem into smaller overlapping regions in order to properly evaluate the integral. This is a piecewise function.
BIBO Systems are stable systems that we value for their predictable and controllable behavior (not to guarantee that they are LTI). Given a bounded input, they produce a bounded output–i.e., they exist in L1 space (or l1 for discrete time). There are several types of stability, including strict, metastable, and unstable.
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