# 3.2 Linear filtering

 Page 1 / 1
Describes signals that cannot be precisely characterized.

## Integration

$Z()=\int_{a}^{b} {X}_{t}()\,d t$

## Linear processing

${Y}_{t}=\int_{()} \,d$ h t X

## Differentiation

${X}_{t}^\prime =\frac{d {X}_{t}}{d t}}$

## Properties

• $\langle Z\rangle =\langle \int_{a}^{b} {X}_{t}()\,d t\rangle =\int_{a}^{b} {}_{X}(t)\,d t$
• $\langle Z^{2}\rangle =\langle \int_{a}^{b} {X}_{{t}_{2}}()\,d {t}_{2}\int_{a}^{b} \overline{{X}_{{t}_{1}}}\,d {t}_{1}\rangle =\int_{a}^{b} \int_{a}^{b} {R}_{X}({t}_{2}, {t}_{1})\,d {t}_{1}\,d {t}_{2}$

${}_{Y}(t)=\langle \int_{()} \,d \rangle$ h t X h t X
If ${X}_{t}$ is wide sense stationary and the linear system is time invariant
${}_{Y}(t)=\int_{()} \,d$ h t X X t h t Y
${R}_{YX}({t}_{2}, {t}_{1})=\langle {Y}_{{t}_{2}}\overline{{X}_{{t}_{1}}}\rangle =\langle \int_{()} \,d \rangle$ h t 2 X X t 1 h t 2 R X t 1
${R}_{YX}({t}_{2}, {t}_{1})=\int_{()} \,d ^\prime$ h t 2 t 1 R X h R X t 2 t 1
where $^\prime =-{t}_{1}$ .
${R}_{Y}({t}_{2}, {t}_{1})=\langle {Y}_{{t}_{2}}\overline{{Y}_{{t}_{1}}}\rangle =\langle {Y}_{{t}_{2}}\int_{()} \,d \rangle$ h t 1 X h t 1 R Y X t 2 h t 1 R Y X t 2
${R}_{Y}({t}_{2}, {t}_{1})=\int_{()} \,d ^\prime$ h t 2 t 1 R Y X R Y t 2 t 1 h ~ R Y X t 2 t 1
where $^\prime ={t}_{2}-$ and $\stackrel{~}{h}()=h(-)$ for all $\in \mathbb{R}$ . ${Y}_{t}$ is WSS if ${X}_{t}$ is WSS and the linear system is time-invariant.

${X}_{t}$ is a wide sense stationary process with ${}_{X}=0$ , and ${R}_{X}()=\frac{{N}_{0}}{2}()$ .Consider the random process going through a filter with impulse response $h(t)=e^{-(at)}u(t)$ .The output process is denoted by ${Y}_{t}$ . ${}_{Y}(t)=0$ for all $t$ .

${R}_{Y}()=\frac{{N}_{0}}{2}\int_{()} \,d$ h h N 0 2 a 2 a
${X}_{t}$ is called a white process. ${Y}_{t}$ is a Markov process.

Power Spectral Density
The power spectral density function of a wide sense stationary (WSS) process ${X}_{t}$ is defined to be the Fourier transform of the autocorrelation functionof ${X}_{t}$ .
${S}_{X}(f)=\int_{()} \,d$ R X 2 f
if ${X}_{t}$ is WSS with autocorrelation function ${R}_{X}()$ .

## Properties

• ${S}_{X}(f)={S}_{X}(-f)$ since ${R}_{X}$ is even and real.
• $\mathrm{Var}({X}_{t})={R}_{X}(0)=\int_{()} \,d f$ S X f
• ${S}_{X}(f)$ is real and nonnegative ${S}_{X}(f)\ge 0$ for all $f$ .

If ${Y}_{t}=\int_{()} \,d$ h t X then

${S}_{Y}(f)=({R}_{Y}())=((h, \stackrel{~}{h}, {R}_{X}()))=H(f)\stackrel{~}{H}(f){S}_{X}(f)=\left|H(f)\right|^{2}{S}_{X}(f)$
since $\stackrel{~}{H}(f)=\int_{()} \,d t$ h ~ t 2 f t H f

${X}_{t}$ is a white process and $h(t)=e^{-(at)}u(t)$ .

$H(f)=\frac{1}{a+i\times 2\pi f}$
${S}_{Y}(f)=\frac{\frac{{N}_{0}}{2}}{a^{2}+4\pi ^{2}f^{2}}$

price elasticity of demand is the degree of responsiveness of a quantity demanded to the change in price of the commodity in question.
what is the importance of learning economics?
it helps to make the correct choice
it helps firm to produce products that will bring more profit
the difference between needs and wants
needs are things that we basically can't live without wants are just luxury things
Thelma
needs are things without them we can't live but want are things without we can live
KP
what is education
KP
it's a process in which we give or receiving methodical instructions
Thelma
what is mixed economy
Amex
who are u?
Lamine
haha
Cleaford
scarm
nura
what it this
Cleaford
hi y'all
Dope
how does group chat help y'all 🤔
Dope
hi y'all
Dope
how does group chat help y'all 🤔
Dope
how does group chat help y'all 🤔
Dope
to learn from one another
Lamine
oh okay
Dope
😟
Creative
Yes
Lamine
what is type of economic
how to understand basics of economics
what is demand schedle
When you make a Scedule of the demand you made
Rodeen
What is macroeconomics
It's one of the two branches of Economics that deal with the aggregate economy.
Mayen
it's about inflation, occupation, gdp and so on
alberto
What is differences between Microeconomics and Macroeconomic?
Bethrand
microeconomics focuses on the action of individual agents in the economy such as businesses, workers and household. while macroeconomics looks at the economy as a whole. it focuses on broad issues in the economy such as government deficit, economy growth, levels of exports and imports, and
Thelma
inflationary increase in prices
Thelma
a price floor of 24 imposed
monopolistic competition
yap
nura
any one there to answer my question
Fixed Costs per week Variable Costs per bear Rent & Rates of Factory Hire & machines Heating & Lighting Repayment of Bank Loan K100.00 K45.00 K5.00 K50.00 Materials Foam Wages K6.00 K1.00 K1.00 Total K200.00 K8.00
Richard
one of the scarce resources that constrain our behaviour is time. each of us has only 24 hours in a day. how do you go about allocating your time in a given day among completing alternatives? once you choose a most important use of time. why do you not spend all your time to it. use the notion of op
mohsina mala..Bangla app hobe na
mani Baba. First learn the spelling of Economics
Economics- The study of how people use their limited resources to tey and satisfy unlimited wants.
Kelly
hmmm
Mani
etar bangla apps hobe na?
Mohsina
Difference between extinct and extici spicies
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
in a comparison of the stages of meiosis to the stage of mitosis, which stages are unique to meiosis and which stages have the same event in botg meiosis and mitosis
Researchers demonstrated that the hippocampus functions in memory processing by creating lesions in the hippocampi of rats, which resulted in ________.
The formulation of new memories is sometimes called ________, and the process of bringing up old memories is called ________.
Got questions? Join the online conversation and get instant answers!