there is an Element of M (an element of R, positive valued) Such that for all time, the absolute value of y(t) is less than M.

An LTI system is BIBO Stable iff:

There exists some M, an element of R, such that the integral of the absolute value of the impulse response over all time is less than M. Essentially, the impulse response must be an element of L1, meaning that it is absolutely integrable. Now a proof:
BIBO if h has an absolute integral.
given h is ai, we have a BIBO System
We most prove that for an arbitrary x(t) (Bounded), y(t) Bounded
The Integral over all time of x(t)*h(t-tao)dtao=abs(y(t))<=abs(x(t))*abs(h(t-tao))dtao=int(B*H)

Principles of proofs:
Iff: If and only if.
Given constraints, we want to choose the optimal choice.

We used this chart to map the notes to frequencies (http://www.phy.mtu.edu/~suits/notefreqs.html) . Afterward, we had to echo the signal through convolution, so we multiplied a decaying exponential by square wave that we raised by its amplitude in order to create a decaying echo. We might have made the echo longer or decay faster, but we like the way it sounds.

SmokeontheWater

http://301.jacpot.us/wp-content/uploads/2009/09/SmokeontheWater.m

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.

Week 2 Summary

One topic covered on Tuesday was energy. Energy says very little about peak magnitude, but both are important consideration. Energy is a norm with p=2, the Integral is normal of absolute value. The Infinite Norm returns the maximum value or the essential supremum. Power is the norm per unit of time. Watts are Joules per second. We discussed different types of signals such as the unit step and the unit pulse. Another function covered was the delta function which is a sifting function which makes it good for taking one time value out of the signal. Next we covered signal decomposition which involves using the delta function to break the signal into pieces that can be worked with. The continuous time delta function presents challenges because we aren’t able to have something that integrates over all time to 1.

The next day we discussed complex sinusoids as spirals in 3D space for a continuous time signal and a sampled version of dots for discrete time. We also discussed how although all continuous time sinusoids are periodic, not all discrete time sinusoids are periodic. The next major theme related plotting complex exponential signals as points in the complex plane with the real part signifying the exponential behavior of the signal and the imaginary part dictating the sinusoidal behavior. For continuous time we use Cartesian coordinates but in discrete time we use polar coordinates because we are only concerned with the –pi to pi range. Finally we discussed how systems are operators that act on signals. There are four different classifications of systems linear, nonlinear, time-invariant, and time-varying.

]]> http://301.jacpot.us/2009/09/3/feed/ 0 http://301.jacpot.us/2009/09/four-ways-of-looking-at-a-signal/ http://301.jacpot.us/2009/09/four-ways-of-looking-at-a-signal/#comments Thu, 03 Sep 2009 09:30:00 +0000 Jacob Poteet http://301.jacpot.us/?p=20 Energy says very little about peak magnitude, but both are important considerations.

Generalized Energy: Norms (Energy is a norm with p=2), Integral is normal of absolute value, Infinite Norm returns the maximum value or the essential supremum.

Power is the norm per unit of time. Watts are Joules per second.

Unit Pulse

Unit Step

Delta Function
A sifting function (good for taking one time value out of the signal)

Signal Decomposition
Using a delta function to break things down.

Delta of T

Continuous time delta function: Presents challenges because we aren’t able to have something the integrates over all time to 1.

What are generalized functions.

CT

Phase Pfrequency (Phase is a frequency dependent time delay).

On Tuesday, Baraniuk emphasized the importance of analyzing learning needs and developing strategies that incorporate them. He also cited evidence that group work improves retention and understanding, and argued that taking notes from a projector is a ‘metacognitive process’ that improves the learning process.

On Thursday, we spent the first 50 minutes of class discussing interesting examples of the use of DSP. The Speak and Spell was a toy from the 1980s that used a TI chip for speech synthesis; this chip fits into a larger history of implementing DSP solutions through specialized hardware. Other products include routers, modems, and music equipment. DSP techniques can eliminate echo and support our massive telecomm infrastructure. DSP allows us to see inside regions we cannot access directly. By using radar and sonar, we can see behind the clouds on Venus and locate oil deposits under the ocean’s floor. We also use signal processing for medical applications including fMRI (Tomography), eye implants, and cochlear implants (http://www.npr.org/templates/story/story.php?storyId=1080910) . We ended our discussion of  human modification just a little before going off the deep end regarding the Singularity. If you’re interested in a TV series about cyborgs, identity, and signals, I recommend Ghost in the Shell.

The foundation of signal processing is an understanding of types of signals themselves. We discussed four types of signals, the combinations of continuous or discrete time with continuous or discreet values. We also drew attention to concepts like periodicity (it’s not that different in discrete time), continuity, sampling and digitization, finite length and zero-padding and windowing.