Signals Important Things to Remember

cos²(t) = ?

$$\cos^2\left(t\right)=\frac{1+\cos\left(2t\right)}{2}$$

Steps to find impulse response of LCCDifferenceE

1. Write in advance form

2. Get constants $b_N$ and $a_N$ from x[n] and y[n] respectively

3. Find roots using characteristic equation of homogeneous part

4. Construct $y_h[n]$ with Cs

5. Write in delay form

6. solve for y[n] on its own on the left

7. Find $y_{}[0]$, $y_{}[1]$, etc (until n = N - 1) using x[n] = $\delta\left\lbrack n\right\rbrack$ and y[n] = h[n] and initial rest

8. Determine C values with determined initial conditions on the homogenous solution

9. plug everything into formula for h[n]

Steps to find impulse response of LCCDifferentialE

1. Find $b_N$ IF N = M

2. Write in terms of Q(D) and P(D)

3. Calculate roots from Q(D) characteristic equation

4. construct y_{h}\left(t\righ)using roots and unknown C_i$values</p></li><li><p>Use standard initial rest conditions of y[0] = 1 OR 0 for 1st or 2nd order and$\frac{dy(0)}{\differentialD t}$= 1 for second order applied to the homogeneous solution to determine$C_i$values</p></li><li><p>plug values of$b_N$,$y_h(t)$$ , and P(D) into formula

What to write for memory/memoryless?

Memoryless means the output at time t depends only on the input at the same instant, t, and not on any input at a time before the current time

What to show to determine if a system is bounded?

Show that if the absolute value of the input is bounded ($|x(t)| < B$) then the output is also bounded ($|y(t)| < 1 + B$)

steps to show linearity

1. Define inputs $x_1(t)$ , $x_2(t)$ , and $x_3(t) = ax_1(t) + bx_2(t)$

2. Plug all three inputs into y(t) to get $y_1(t), y_2(t), y_3(t)$

3. Check for $y_3(t) = ay_1(t) + by_2(t)$

   1. True = Linear

   2. False = Not Linear

Steps to show time-invariance

1. Time delay the input - $x\left(t\right)\longrightarrow{}x\left(t-t_0\right)$

   1. i.e. replace only the t in x(t) with $t - t_0$

2. Time delay the original output - $y\left(t\right)\longrightarrow{}y\left(t-t_0\right)$

   1. i.e. replace all $t\longrightarrow{}t-t_0$

3. Compare to each other

   1. if equal, time invariant

   2. if not equal, not time invariant

What to write for causal?

The output at time t depends only on inputs at or before time t, and not on inputs at times after t (future times)

What is convolution? (description)

Convolution is a mathematical method for determining the output of a system by taking the integral or sum of the input combined with the system’s impulse response. It essentially breaks down a complex input into many shifted and scaled impulses to determine the output with any arbitrary input (not just delta impulse as the input).

The system must be:

• linear

• time invariant

• causal

How to show if is system is invertible?

1. solve for x[n] explicitly

2. verify that the solution holds for ALL n

How to perform CT convolution of two functions with GRAPHICAL METHOD

1. graph both functions separately

2. choose one for $t\longrightarrow{}\tau$ and one for $t\longrightarrow{}-\tau+t$

3. graph new functions with respect to Tau

4. combine the two into one graph by sliding the $x_{i}\left(t-\tau)\right.$ function across x($\tau$)

5. use the definition of convolution to find the solutions for all significant intervals by integration of the multiplication of the two function with respect to Tau

   1. The definition being $x_1\left(t\right)\ast x_2\left(t\right)=\int_0^{\infty}\!x_1\left(\tau\right)\cdot x_2\left(t-\tau\right)\,d\tau$

How to perform CT convolution with the table

1. Use properties of convolution to organize the convolution of the two functions in a form that is on the table

   1. linearity allows for convolution distribution

   2. time shifting of the functions is allowed

2. Substitute the solution from the table into the $x_1\left(t\right)\ast x_2\left(t\right)$ equation

How to perform CT convolution with just the definition

1. Identify one function as $x\left(t\right)\longrightarrow{}x\left(\tau\right)$ and one as $x_2\left(t\right)\longrightarrow{}x\left(t-\tau\right)_{}$

2. Multiply the functions and perform the integration as the definition of convolution describes

How to perform DT convolution with the graphical method

1. plot both functions

2. set one function to be x[k] and one to be h[k]

3. flip h[k] ==> h[-k]

4. shift h[-k] by n ==> h[n - k]

5. plot the new functions in terms of k

6. multiply overlapping values

7. sum overlapping values to get y[n]

8. repeat for each relevant value of n

9. plot the final y[n]

How to perform DT convolution with the definition

1. set one function to be x[k] and one to be h[k]

2. flip h[k] ==> h[-k]

3. shift h[-k] by n ==> h[n - k]

4. substitute the two functions into the definition of DT convolution

5. identify relevant interval of k for which the product of both functions is non-zero, ignore the rest of sum over k

6. sum the product of the functions over k

7. repeat for each relevant value of n