In our study of discrete-time signals and signal processing, there are five very important signals that we will use to both illustrate signal processing concepts, and also to probe or test signal processing systems: the
delta function , the
unit step function , the
unit pulse function , the
real exponential function ,
sinusoidal functions , and
complex exponential functions . This module will consider the first four; sinusoids and complex exponentials are particularly important, so a separate model will cover them. Each of these signals will be introduced as infinite-length signals, but they all have straightforward finite-length equivalents.
The discrete-time delta function
The delta function is probably the simplest nontrivial signal. It is represented mathematically with (no surprise) the Greek letter delta: $\delta[n]$. It takes the value 0 for all time points, except at the time point 0 where it peaks up to the value 1:$\delta[n]=\begin{cases}1&n=0 \\ 0&\textrm{otherwise}\end{cases}$
The discrete-time delta function. In a variety of important settings, we will often see the delta function shifted by a particular time value. The delta function $\delta[n-m]$ is 0, except for a peak of 1 at time $m$:
A time-shifted discrete-time delta function $\delta[n-m]$, where $m=9$. One of the reasons the shifted delta function is so useful is that we can use it to select, or sample, a value of another signal at some defined time value. Suppose we have some signal $x[n]$, and we would like to isolate that signal's value at time $m$. What we can do is multiply that signal by a shifted delta signal. We can say $y[n]=x[n]\delta[n-m]$, but since that $y[n]$ will be zero for all $n$ except at $n=m$, it is equivalent to express it as $y[n]=x[m]\delta[n-m]$, where now $x[m]$ is no longer a function, but a constant. The following figure shows how this operation isolates a particular time sample of $x[n]$:
Using a time-shifted delta function to isolate a sample of the signal $x[n]$.
The unit step function
The unit step function can be thought of like turning on a switch. Usually identified as $u[n]$, it is $0$ for all $n \lt 0$, and then at $n=0$ it "switches on" and is $1$ for all $n \geq 0$: $u[n]=\begin{cases}1&n \lt 0\\ 1&n\geq 0\end{cases}$:
The unit step function. As with the delta function, it will also be useful for us to shift the step function:
A shifted step function $[n-m]$ with $m=5$. And, as you might have guessed, we can use a shifted step function in a similar way to the delta function by multiplying it with another signal. Whereas the delta function selected a single value of a certain signal (zeroing out the rest), the step function isolates a portion of a signal after a given time. Below, a step function is used to zero out all the values of $x[n]$ for $n\lt 5$, keeping the rest:
A shifted step function can be used to zero out all values of a signal before a certain time index. Supposing a signal $x[n]$ were not causal, setting $m$ to zero and performing the operation $x[n]u[n]$ would zero out all values of $x[n]$ before $n=0$, thereby making the result causal.
The unit pulse function
The unit pulse $p[n]$ is very similar to the unit step function in how it "switches on" from 0 to 1, but then it also "switches off" at a later time. We will say it "switches on" at time $N_1$, and "off" at time $N_2$: $p[n] = \begin{cases}0&n\lt N_1 \\
1&N_1 \leq n \leq N_2 \\
0&n\gt N_2\\
\end{cases}$
The unit pulse function $p[n]$, here with $N_1 = −5$ and $N_2= 3$. Of course, rather than use the above piece-wise notation, it is also possible to express the pulse as the difference of two step functions: $p[n] = u[n-N_1]- u[n-(N_2+1)]$.
The real exponential function
Finally, we have the real exponential function, which takes a real number $a$ (that we are going to assume is positive) and raises it to the power of $n,$ where $n$ is the time index: $r[n] = a^n$, $a\in R$, $a\geq 0$. So at $n=0$, $r[n]=a^0$, at $n=1$ it equals $a$, is $a^2$ at $n=2$, and so on. As the name suggests, the signal will exponentially increase or decrease, depending on the value of $a$.
For $a\gt 1$, the real exponential function increases with time. Here $a=1.1$.For $0 \lt a\lt 1$, the real exponential function decreases with time (or we could say it increases exponentially as the time index decreases). Here $a=.9$.
Questions & Answers
differentiate between demand and supply
giving examples
In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,
When MP₁ becomes negative, TP start to decline.
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab
Kelo
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)
Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded
Ezea
ok
Shukri
how do you save a country economic situation when it's falling apart
Economic growth as an increase in the production and consumption of goods and services within an economy.but
Economic development as a broader concept that encompasses not only economic growth but also social & human well being.
Shukri
production function means
Jabir
What do you think is more important to focus on when considering inequality ?
sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏
Asui
it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs
Awais
thank you so much 👍 sir
Asui
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p
Cornelius
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities,
Cornelius
Suppose a consumer consuming two commodities X and Y has
The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50.
A,Calculate quantities of x and y which maximize utility.
B,Calculate value of Lagrange multiplier.
C,Calculate quantities of X and Y consumed with a given price.
D,alculate optimum level of output .
the market for lemon has 10 potential consumers, each having an individual demand curve p=101-10Qi, where p is price in dollar's per cup and Qi is the number of cups demanded per week by the i th consumer.Find the market demand curve using algebra. Draw an individual demand curve and the market dema
suppose the production function is given by ( L, K)=L¼K¾.assuming capital is fixed find APL and MPL. consider the following short run production function:Q=6L²-0.4L³ a) find the value of L that maximizes output b)find the value of L that maximizes marginal product
