1. Signals and Systems
EE235
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2. Today’s menu Homework 2 due LTI System – Impulse response
Lead in to Convolution
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3. Impulse response (Definition)
Any signal can be built out of impulses
Impulse response is the response of any Linear Time Invariant system when the input is a unit impulse
Impulse Response h(t)
Definition. h(t) is the response of the system when the input is given a Dirac Delta function.
Delta(t) into the box, out comes h(t) (which is just another function in terms of time).
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4. Briefly: recall superposition
Superposition is…
Weighted sum of inputs  weighted sum of outputs
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5. Building x(t) with δ(t)
Using the sifting properties:
Change of variable:
t  t
t0  t
From a constant to a variable
Last line: we can do the -1 inside the parathesis because mathematically they are identical (the delta function is still spiking at t=tau). It also only shifts the direction of “t” later on when we do the “shift” in convolution. =
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6. Building x(t) with δ(t)
Jumped a few steps…
In the previous slide, we also jumped a few steps and here they are. (Go through the math).
What does this physically mean, though? An integral is basically an infinite sum of something. This something is the product of the function x (in tau axis) and a delta function.
This is a significant result because x(t) is now an infinite sum of delta functions. It is a strange way of seeing the same thing, but you will see the significance later.
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7. Building x(t) with δ(t)
Another way to see…
x(t) t
Value at the “tip”
dD(t) t D
1/D
Compensate for the height of the “unit pulse”
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8. So what? Two things we have learned
If the system is LTI, we can completely characterize the system by how it responds to an input impulse.
Impulse Response h(t)
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9. Shifted Impulse  Shifted Impulse response
h(t)
For LTI system T
x(t)
y(t) T
d(t)
h(t)
Impulse  Impulse response
Summary of the definitions: T is the “transformation” of the signal x(t) and turning it into y(t). T is the “system”, characterized with a continuous time line.
And h(t) is the output (like the y(t) if the system T has an input delta(t)). Learn this by heart and understand it by heart. T
d(t-t0)
h(t-t0)
Shifted Impulse  Shifted Impulse response
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10. Finding Impulse Response (examples)
Let x(t)=d(t)
What is h(t)?
First one is a delay, second one just an amplification, and the third one is an integral;
Integrating delta(t) you get u(t) (observe the limit).
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11. Finding Impulse Response
For an LTI system, if
x(t)=d(t-1)  y(t)=u(t)-u(t-2)
What is h(t)?
Remember the definition, and that this is time invariant
h(t)
d(t-1)
u(t)-u(t-2)
h(t)=u(t+1)-u(t-1)
An impulse turns into two unit steps shifted in time
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