One of the most general ways of defining an LTI system is by expressing its output at some time $n$ as a weighted sum of values of the input and output at a finite number various other times. If all those times are at or prior to $n$, such systems a system is causal. In terms of a difference equation, the system definition would look something like this:
$y[n]~=~ b_0 x[n] + b_1 x[n-1]+ b_2 x[n-2] + \cdots + b_M x[n-M]-a_1 y[n-1] - a_2 y[n-2]+ \cdots - a_N y[n-N]$This kind of system can also be displayed as a block diagram:
CAPTION. In the block diagram, the $z^{-1}$ blocks represent time delays, the triangles represent multiplications, and the $\Sigma$ blocks represent summations of the signals being input into them. The output at time $n$ is a weighted sum of the previous $M$ input values (along with the current value) and previous $N$ output values.
Lti systems in the z-domain
Consider again the input/output relationship of a causal LTI system:
$y[n]~=~ b_0 x[n] + b_1 x[n-1]+ b_2 x[n-2] + \cdots + b_M x[n-M]-a_1 y[n-1] - a_2 y[n-2]+ \cdots - a_N y[n-N]$We can take the z-transform of this whole equation, bearing in mind that the transform is a linear operation and that delays of $n_0$ in time correspond to multiplications by $z^{-n_0}$ in the z-domain. That would leave us with this:
$Y(z)=~ b_0 X(z) + b_1 z^{-1} X(z) + b_2 z^{-2} X(z) + \cdots b_M z^{-M} X(z)- a_1 z^{-1} Y(z) - a_2 z^{-2} Y(z) - \cdots - a_N z^{-N} Y(z)$From there we can do a little rearranging:
$\begin{align*}Y(z)&=b_0 X(z) + b_1 z^{-1} X(z) + b_2 z^{-2} X(z) + \cdots b_M z^{-M} X(z)- a_1 z^{-1} Y(z) - a_2 z^{-2} Y(z) - \cdots - a_N z^{-N} Y(z)\\
Y(z)+ a_1 z^{-1} Y(z) a_2 z^{-2} Y(z) + \cdots + a_N z^{-N} Y(z)&=b_0 X(z) + b_1 z^{-1} X(z) + b_2 z^{-2} X(z) + \cdots b_M z^{-M}X(z)\\
Y(z)(1+a_1 z^{-1} + a_2 z^{-2} + \cdots + a_Nz^{-N})&=X(z)(b_0 + b_1 z^{-1} + b_2 z^{-2} + \cdots b_M z^{-M})\\
H(z)=\frac{Y(z)}{X(z)}&=\frac{b_0 + b_1 z^{-1} + b_2 z^{-2} + \cdots b_M z^{-M}}{1+a_1 z^{-1} + a_2 z^{-2} + \cdots + a_Nz^{-N}}
\end{align*}$This $H(z)$, which is a rational function in terms of $z$, is called the system's
transfer function .
The transfer function: poles and zeroes, roc, and stability
So we see that the transfer function for one of the most common classes of LTI systems--in which the output is a weighted sum of a finite number of past inputs and outputs--is a rational function in terms of $z$:
$H(z)=\frac{b_0 + b_1 z^{-1} + b_2 z^{-2} + \cdots b_M z^{-M}}{1+a_1 z^{-1} + a_2 z^{-2} + \cdots + a_Nz^{-N}}$We can express this function in terms of positive powers of $z$ by simple factoring, and then according to the fundamental theorem of algebra, we can represent the numerator and denominator polynomials with respect to their roots:
$\begin{align*}H(z)&=\frac{b_0 + b_1 z^{-1} + b_2 z^{-2} + \cdots b_M z^{-M}}{1+a_1 z^{-1} + a_2 z^{-2} + \cdots + a_Nz^{-N}}\\&=\frac{z^{-M}(b_0z^M+b_1z^{M-1}+b_2z^{M-2}+\cdots+b_M)}{z^{-N}(z^N+a_1z^{N-1}+a_2z^{N-2}+\cdots+a_N)}\\&=\frac{z^{-M}b_0(z-\zeta_1)(z-\zeta_2)\cdots(z-\zeta_M)}{z^{-N}(z-p_1)(z-p_2)\cdots(z-p_N)}
\end{align*}$The values of $z$ for which the numerator is zero are called
zeros , and the values for which the denominator is zero are called
poles .
Communication is effective because it allows individuals to share ideas, thoughts, and information with others.
effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.
it starts up serve and return practice/assessments.it helps find voice talking therapy also assessments through relaxed conversation.
miss
Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.
Samuel
I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills
