March 13 - 20 Flashcards
DT LTI Systems: what is that
Systems that are both linear and satisfy superposition, and time-invariant (same response for shifted inputs)
FIR vs IIR Systems
Finite impulse response: Impulse response h[n] is non-zero for a finite number of steps
Infinite impulse response: impulse response h[n] extends infinitely
What is convolution
Convolution allows us to compute the output y[n] of an LTI system given an input u[n] and impulse response h[n]
This means: System scales each input u[k] by the impulse response value h[n-k]
Then all scaled responses are added together
How convolution works
flip h[k] to get h[-k]
shift h[-k] by n to get h[n-k]
multiply u[k] and h[n-k] for all k
sum up the values
Difference eqns : what is it
Difference eqn describes how an output y[n] depends on past values of itself and past/current values of the input u[n].
System is non-recursive if coeff ai = o for all i. (only depends on inputs)
If at least one ai != 0, system is recursive (depends on past outputs)
Comparing Convolution vs
Convolution : output depends on all past and current inputs, computation is heavy, works for all LTI systems
Diff eqn: Outputs depends on past outputs and inputs, less computation heavy, only works for specific cases
If both can be applied, diff eqn is preferred due to computation ease
20-point moving average example
Moving average filter smooths a signal by averaging over past samples.
Standard non recursive diff eqn sums 20 terms for every n given.
A recursive form avoids summing 20 terms for every n.
DT LTI system block diagram
every DT LTI with finite memory (FIR) can be built using:
1. Multipliers: scale a signal
2. Adders: Sum multiple signals
3. Unit Delays: Store past values of signals
Intro to State space (SS) Eqns
State space eqns are a way to represent DT LTI systems using first-order difference eqns. Rather than dealing with higher-order diff eqns, we break them into a system of first-order eqns.
Consider y[n] = 3u[n] - 2u[n-1] +5u[n-3]
To express this in state space form, we define state vars:
x1[n+1] = u[n]
x2[n+1] = x1[n]
x3[n+1] = x2[n]
Describes the system in terms of state vars
Matrix representation of state-space eqns
We can express system in matrix form
x[n+1] = Ax[n] + Bu[n]
y[n] = Cx[n] + Du[n]
A reps state transition matrix
B reps input matrix
C reps output matrix
D reps direct transmission term
Why use state space eqns
Compact representation of higher order systems
Easier for computers to simulate real-time systems
Useful for control system eg. robotics, digital filters
Computing System response using SS eqns
to compute the output y[n], we need
initial state x[0]
input u[n] for all n.
System evolves recursively using: matrix form eqn.
Method is used for real-time DSP applications such as digital filters
The Z-Transform
Z transform is used to analyze discrete time signals and difference equations. There are two types:
Unilateral Z-transform: used for casual systems where x[n] = 0 for n < 0, helps solve diff eqn with initial conditions
Bilateral Z-transform: used for both casual and non-casual systems. Useful for frequency-domain analysis
Computing the inverse Z-Transform
To find x[n] we use
1. Partial Fraction Expansion: if X(z) is rational, split into simpler terms.
2. Power series expansion
3. Residue Method: use contour integration
