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Stochastic process | Stochastic process | ||
− | The idea of a stochastic process is a straightforward extension of that of random variable. Instead of mapping each <math>\omega\in\mathcal{S}</math> of a random experiment to a number <math>\mathbf{X}\left(\omega\right)</math> , we map it to a function of time <math>\mathbf{X}\left(t,\omega\right)</math> that is called sample function. | + | The idea of a stochastic process is a straightforward extension of that of random variable. Instead of mapping each <math class="inline">\omega\in\mathcal{S}</math> of a random experiment to a number <math class="inline">\mathbf{X}\left(\omega\right)</math> , we map it to a function of time <math class="inline">\mathbf{X}\left(t,\omega\right)</math> that is called sample function. |
Note | Note | ||
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Note | Note | ||
− | If we pick a particular point in time <math>t=t_{1}</math> , we have <math>\mathbf{X}\left(t_{1},\omega\right)</math> that is a random variable. | + | If we pick a particular point in time <math class="inline">t=t_{1}</math> , we have <math class="inline">\mathbf{X}\left(t_{1},\omega\right)</math> that is a random variable. |
Definition | Definition | ||
− | A stochastic process or random process defined on <math>\left(\mathcal{S},\mathcal{F},\mathcal{P}\right)</math> is a famility of random variables | + | A stochastic process or random process defined on <math class="inline">\left(\mathcal{S},\mathcal{F},\mathcal{P}\right)</math> is a famility of random variables |
− | <math>\left\{ \mathbf{X}\left(t\right):t\in\mathbf{T}\right\}</math> | + | <math class="inline">\left\{ \mathbf{X}\left(t\right):t\in\mathbf{T}\right\}</math> |
− | indexed by <math>t</math> , where the index set <math>\mathbf{T}</math> can be discrete or continuous. | + | indexed by <math class="inline">t</math> , where the index set <math class="inline">\mathbf{T}</math> can be discrete or continuous. |
Note | Note | ||
− | 1. If <math>\mathbf{T}</math> is an uncountable subset of <math>\mathbf{R}</math> , <math>\mathbf{X}\left(t\right)</math> is called a continuous-time random process. | + | 1. If <math class="inline">\mathbf{T}</math> is an uncountable subset of <math class="inline">\mathbf{R}</math> , <math class="inline">\mathbf{X}\left(t\right)</math> is called a continuous-time random process. |
− | 2. If <math>\mathbf{T}</math> is an discrete set, <math>\mathbf{X}\left(t\right)</math> is called a discrete-time random process. | + | 2. If <math class="inline">\mathbf{T}</math> is an discrete set, <math class="inline">\mathbf{X}\left(t\right)</math> is called a discrete-time random process. |
− | 3. <math>\mathbf{X}\left(t\right)</math> is called a discrete-state random process if for all <math>t\in\mathbf{T}</math> , it takes on values from a discrete set. Otherwise it is called a continuous-state random process. | + | 3. <math class="inline">\mathbf{X}\left(t\right)</math> is called a discrete-state random process if for all <math class="inline">t\in\mathbf{T}</math> , it takes on values from a discrete set. Otherwise it is called a continuous-state random process. |
Note | Note | ||
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Notation | Notation | ||
− | We will use the notation <math>\mathbf{X}\left(t\right)</math> to represent a random process (just as we use <math>\mathbf{X}</math> to represent a random variable). Technically we should write <math>\mathbf{X}\left(t,\omega\right)</math> or <math>\mathbf{X}\left(\cdot,\cdot\right)</math> (just as technically we should write <math>\mathbf{X}\left(\omega\right)</math> or <math>\mathbf{X}\left(\cdot\right)</math> for a random variable). | + | We will use the notation <math class="inline">\mathbf{X}\left(t\right)</math> to represent a random process (just as we use <math class="inline">\mathbf{X}</math> to represent a random variable). Technically we should write <math class="inline">\mathbf{X}\left(t,\omega\right)</math> or <math class="inline">\mathbf{X}\left(\cdot,\cdot\right)</math> (just as technically we should write <math class="inline">\mathbf{X}\left(\omega\right)</math> or <math class="inline">\mathbf{X}\left(\cdot\right)</math> for a random variable). |
− | <math>\mathbf{X}\left(\cdot,\cdot\right):\mathbf{T}\times\mathcal{S}\longrightarrow\mathbf{R}.</math> | + | <math class="inline">\mathbf{X}\left(\cdot,\cdot\right):\mathbf{T}\times\mathcal{S}\longrightarrow\mathbf{R}.</math> |
note | note | ||
− | • <math>\mathbf{X}\left(t,\omega\right)</math> or <math>\mathbf{X}\left(\cdot,\cdot\right)</math> is a random process. | + | • <math class="inline">\mathbf{X}\left(t,\omega\right)</math> or <math class="inline">\mathbf{X}\left(\cdot,\cdot\right)</math> is a random process. |
− | • <math>\mathbf{X}\left(t_{0},\omega\right)</math> or <math>\mathbf{X}\left(t_{0},\cdot\right)</math> is a random variable for fixed </math>t_{0}</math> . | + | • <math class="inline">\mathbf{X}\left(t_{0},\omega\right)</math> or <math class="inline">\mathbf{X}\left(t_{0},\cdot\right)</math> is a random variable for fixed </math>t_{0}</math> . |
− | • <math>\mathbf{X}\left(t,\omega_{0}\right)</math> or <math>\mathbf{X}\left(\cdot,\omega_{0}\right)</math> for a fixed <math>\omega_{0}\in\mathcal{S}</math> is a function of time or sample function. | + | • <math class="inline">\mathbf{X}\left(t,\omega_{0}\right)</math> or <math class="inline">\mathbf{X}\left(\cdot,\omega_{0}\right)</math> for a fixed <math class="inline">\omega_{0}\in\mathcal{S}</math> is a function of time or sample function. |
− | • <math>\mathbf{X}\left(t_{0},\omega_{0}\right)</math> is a real number. | + | • <math class="inline">\mathbf{X}\left(t_{0},\omega_{0}\right)</math> is a real number. |
3.1.1 Statistics of Stochastic Processes | 3.1.1 Statistics of Stochastic Processes | ||
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Definition | Definition | ||
− | The first-order CDF of a random process <math>\mathbf{X}\left(t\right)</math> is | + | The first-order CDF of a random process <math class="inline">\mathbf{X}\left(t\right)</math> is |
− | <math>F_{\mathbf{X}\left(t\right)}\left(x\right)\triangleq P\left(\left\{ X\left(t\right)\leq x\right\} \right)=F_{\mathbf{X}\left(t\right)}\left(x,t\right)</math>. | + | <math class="inline">F_{\mathbf{X}\left(t\right)}\left(x\right)\triangleq P\left(\left\{ X\left(t\right)\leq x\right\} \right)=F_{\mathbf{X}\left(t\right)}\left(x,t\right)</math>. |
− | The first-order PDF of a random process <math>\mathbf{X}\left(t\right)</math> is | + | The first-order PDF of a random process <math class="inline">\mathbf{X}\left(t\right)</math> is |
− | <math>f_{\mathbf{X}\left(t\right)}\left(x\right)=\frac{dF_{\mathbf{X}\left(t\right)}\left(x\right)}{dx}=f_{\mathbf{X}\left(t\right)}\left(x,t\right).</math> | + | <math class="inline">f_{\mathbf{X}\left(t\right)}\left(x\right)=\frac{dF_{\mathbf{X}\left(t\right)}\left(x\right)}{dx}=f_{\mathbf{X}\left(t\right)}\left(x,t\right).</math> |
The n-th order CDF and PDF | The n-th order CDF and PDF | ||
− | We are often interested in the joint behavior of <math>\mathbf{X}\left(t\right)</math> sample at more than one point in time. First-order description is not sufficient. | + | We are often interested in the joint behavior of <math class="inline">\mathbf{X}\left(t\right)</math> sample at more than one point in time. First-order description is not sufficient. |
Definition | Definition | ||
− | The n-th order CDF of a random process <math>\mathbf{X}\left(t\right)</math> is | + | The n-th order CDF of a random process <math class="inline">\mathbf{X}\left(t\right)</math> is |
− | <math>F_{\mathbf{X}\left(t_{1}\right),\cdots,\mathbf{X}\left(t_{n}\right)}\left(x_{1},\cdots,x_{n}\right)</math> | + | <math class="inline">F_{\mathbf{X}\left(t_{1}\right),\cdots,\mathbf{X}\left(t_{n}\right)}\left(x_{1},\cdots,x_{n}\right)</math> |
and the n-th order PDF is | and the n-th order PDF is | ||
− | <math>f_{\mathbf{X}\left(t_{1}\right),\cdots,\mathbf{X}\left(t_{n}\right)}\left(x_{1},\cdots,x_{n}\right)=\frac{\partial^{n}F_{\mathbf{X}\left(t_{1}\right),\cdots,\mathbf{X}\left(t_{n}\right)}\left(x_{1},\cdots,x_{n}\right)}{\partial x_{1}\cdots\partial x_{n}}.</math> | + | <math class="inline">f_{\mathbf{X}\left(t_{1}\right),\cdots,\mathbf{X}\left(t_{n}\right)}\left(x_{1},\cdots,x_{n}\right)=\frac{\partial^{n}F_{\mathbf{X}\left(t_{1}\right),\cdots,\mathbf{X}\left(t_{n}\right)}\left(x_{1},\cdots,x_{n}\right)}{\partial x_{1}\cdots\partial x_{n}}.</math> |
− | Of particular interest are the second order <math>\left(n=2\right)</math> CDFs and PDFs, <math>F_{\mathbf{X}\left(t_{1}\right),\mathbf{X}\left(t_{2}\right)}\left(x_{1},x_{2};t_{1},t_{2}\right)</math> and <math>f_{\mathbf{X}\left(t_{1}\right),\mathbf{X}\left(t_{2}\right)}\left(x_{1},x_{2};t_{1},t_{2}\right)</math> . | + | Of particular interest are the second order <math class="inline">\left(n=2\right)</math> CDFs and PDFs, <math class="inline">F_{\mathbf{X}\left(t_{1}\right),\mathbf{X}\left(t_{2}\right)}\left(x_{1},x_{2};t_{1},t_{2}\right)</math> and <math class="inline">f_{\mathbf{X}\left(t_{1}\right),\mathbf{X}\left(t_{2}\right)}\left(x_{1},x_{2};t_{1},t_{2}\right)</math> . |
3.1.2 General Properties | 3.1.2 General Properties | ||
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Fact | Fact | ||
− | A complex random process <math>\mathbf{Z}\left(t\right)=\mathbf{X}\left(t\right)+i\mathbf{Y}\left(t\right)</math> where <math>\mathbf{X}\left(t\right)</math> and <math>\mathbf{Y}\left(t\right)</math> are real jointly-distributed random processes, is completely characterized by | + | A complex random process <math class="inline">\mathbf{Z}\left(t\right)=\mathbf{X}\left(t\right)+i\mathbf{Y}\left(t\right)</math> where <math class="inline">\mathbf{X}\left(t\right)</math> and <math class="inline">\mathbf{Y}\left(t\right)</math> are real jointly-distributed random processes, is completely characterized by |
− | <math>F_{\mathbf{X}\left(t_{1}\right),\cdots,\mathbf{X}\left(t_{n}\right),\mathbf{Y}\left(t_{1}\right),\cdots,\mathbf{Y}\left(t_{n}\right)}\left(x_{1},\cdots,x_{n},y_{1},\cdots,y_{n},t_{1},\cdots,t_{n}\right)</math> | + | <math class="inline">F_{\mathbf{X}\left(t_{1}\right),\cdots,\mathbf{X}\left(t_{n}\right),\mathbf{Y}\left(t_{1}\right),\cdots,\mathbf{Y}\left(t_{n}\right)}\left(x_{1},\cdots,x_{n},y_{1},\cdots,y_{n},t_{1},\cdots,t_{n}\right)</math> |
− | for all <math>n\in\mathbf{N}</math> and all <math>t_{1},\cdots,t_{n}</math> . (This is a very hard result to proof.) | + | for all <math class="inline">n\in\mathbf{N}</math> and all <math class="inline">t_{1},\cdots,t_{n}</math> . (This is a very hard result to proof.) |
Fact. The real version | Fact. The real version | ||
− | A real random process <math>\mathbf{X}\left(t\right)</math> is completely chracterized by the joint CDF | + | A real random process <math class="inline">\mathbf{X}\left(t\right)</math> is completely chracterized by the joint CDF |
− | <math>F_{\mathbf{X}\left(t_{1}\right),\cdots,\mathbf{X}\left(t_{n}\right)}\left(x_{1},\cdots,x_{n};t_{1},\cdots,t_{n}\right)</math> | + | <math class="inline">F_{\mathbf{X}\left(t_{1}\right),\cdots,\mathbf{X}\left(t_{n}\right)}\left(x_{1},\cdots,x_{n};t_{1},\cdots,t_{n}\right)</math> |
− | for all <math>n\in\mathbf{N}</math> and all <math>t_{1},\cdots,t_{n}</math> . | + | for all <math class="inline">n\in\mathbf{N}</math> and all <math class="inline">t_{1},\cdots,t_{n}</math> . |
Defintion. Autocorrelation function | Defintion. Autocorrelation function | ||
− | For a random process <math>\mathbf{X}\left(t\right)</math> , real or complex, the autocorrelation function of <math>\mathbf{X}\left(t\right)</math> is | + | For a random process <math class="inline">\mathbf{X}\left(t\right)</math> , real or complex, the autocorrelation function of <math class="inline">\mathbf{X}\left(t\right)</math> is |
− | <math>R_{\mathbf{XX}}\left(t_{1},t_{2}\right)\triangleq E\left[\mathbf{X}\left(t_{1}\right)\mathbf{X}^{*}\left(t_{2}\right)\right].</math> | + | <math class="inline">R_{\mathbf{XX}}\left(t_{1},t_{2}\right)\triangleq E\left[\mathbf{X}\left(t_{1}\right)\mathbf{X}^{*}\left(t_{2}\right)\right].</math> |
Note | Note | ||
− | <math>R_{\mathbf{XX}}\left(t_{1},t_{2}\right)</math> | + | <math class="inline">R_{\mathbf{XX}}\left(t_{1},t_{2}\right)</math> |
Note | Note | ||
− | For <math>\mathbf{X}\left(t\right)</math> real | + | For <math class="inline">\mathbf{X}\left(t\right)</math> real |
− | <math>R_{\mathbf{XX}}\left(t_{1},t_{2}\right)=E\left[\mathbf{X}\left(t_{1}\right)\mathbf{X}\left(t_{2}\right)\right]</math> | + | <math class="inline">R_{\mathbf{XX}}\left(t_{1},t_{2}\right)=E\left[\mathbf{X}\left(t_{1}\right)\mathbf{X}\left(t_{2}\right)\right]</math> |
and | and | ||
− | <math>R_{\mathbf{XX}}\left(t_{1},t_{2}\right)=R_{\mathbf{XX}}\left(t_{2},t_{1}\right).</math> | + | <math class="inline">R_{\mathbf{XX}}\left(t_{1},t_{2}\right)=R_{\mathbf{XX}}\left(t_{2},t_{1}\right).</math> |
Note | Note | ||
− | <math>E\left[\left|\mathbf{X}\left(t\right)\right|^{2}\right]=E\left[\mathbf{X}\left(t\right)\mathbf{X}^{*}\left(t\right)\right]=R_{\mathbf{XX}}\left(t,t\right).</math> | + | <math class="inline">E\left[\left|\mathbf{X}\left(t\right)\right|^{2}\right]=E\left[\mathbf{X}\left(t\right)\mathbf{X}^{*}\left(t\right)\right]=R_{\mathbf{XX}}\left(t,t\right).</math> |
− | <math>E\left[\left|\mathbf{X}\left(t\right)\right|^{2}\right]\geq0\Longrightarrow R_{\mathbf{XX}}\left(t,t\right)\geq0,\quad\forall t.</math> | + | <math class="inline">E\left[\left|\mathbf{X}\left(t\right)\right|^{2}\right]\geq0\Longrightarrow R_{\mathbf{XX}}\left(t,t\right)\geq0,\quad\forall t.</math> |
Definition. Mean | Definition. Mean | ||
− | The mean of a random process <math>\mathbf{X}\left(t\right)</math> is | + | The mean of a random process <math class="inline">\mathbf{X}\left(t\right)</math> is |
− | <math>\eta_{\mathbf{X}}\left(t\right)=E\left[\mathbf{X}\left(t\right)\right].</math> | + | <math class="inline">\eta_{\mathbf{X}}\left(t\right)=E\left[\mathbf{X}\left(t\right)\right].</math> |
− | If <math>\mathbf{X}\left(t\right)</math> is a complex random process with <math>\mathbf{X}\left(t\right)=\mathbf{X}_{R}\left(t\right)+i\mathbf{X}_{I}\left(t\right)</math> , then <math>E\left[\mathbf{X}\left(t\right)\right]=E\left[\mathbf{X}\left(t\right)=\mathbf{X}_{R}\left(t\right)+i\mathbf{X}_{I}\left(t\right)\right]=E\left[\mathbf{X}_{R}\left(t\right)\right]+iE\left[\mathbf{X}_{I}\left(t\right)\right].</math> | + | If <math class="inline">\mathbf{X}\left(t\right)</math> is a complex random process with <math class="inline">\mathbf{X}\left(t\right)=\mathbf{X}_{R}\left(t\right)+i\mathbf{X}_{I}\left(t\right)</math> , then <math class="inline">E\left[\mathbf{X}\left(t\right)\right]=E\left[\mathbf{X}\left(t\right)=\mathbf{X}_{R}\left(t\right)+i\mathbf{X}_{I}\left(t\right)\right]=E\left[\mathbf{X}_{R}\left(t\right)\right]+iE\left[\mathbf{X}_{I}\left(t\right)\right].</math> |
Definition. Autocovariance function | Definition. Autocovariance function | ||
− | The autocovariance function of a random process <math>\mathbf{X}\left(t\right)</math> with mean <math>\eta_{\mathbf{X}}\left(t\right)=E\left[\mathbf{X}\left(t\right)\right]</math> is | + | The autocovariance function of a random process <math class="inline">\mathbf{X}\left(t\right)</math> with mean <math class="inline">\eta_{\mathbf{X}}\left(t\right)=E\left[\mathbf{X}\left(t\right)\right]</math> is |
− | <math>C_{\mathbf{XX}}\left(t_{1},t_{2}\right)</math> | + | <math class="inline">C_{\mathbf{XX}}\left(t_{1},t_{2}\right)</math> |
Fact | Fact | ||
− | The autocorrelation function <math>R_{\mathbf{XX}}\left(t_{1},t_{2}\right)</math> is a non-negative definite function: for any set of numbers <math>\left\{ a_{i}\right\}</math> and times <math>\left\{ t_{i}\right\}</math> , and any <math>n\in\mathbf{N}</math> | + | The autocorrelation function <math class="inline">R_{\mathbf{XX}}\left(t_{1},t_{2}\right)</math> is a non-negative definite function: for any set of numbers <math class="inline">\left\{ a_{i}\right\}</math> and times <math class="inline">\left\{ t_{i}\right\}</math> , and any <math class="inline">n\in\mathbf{N}</math> |
− | <math>\sum_{i=1}^{n}\sum_{j=1}^{n}a_{i}a_{j}^{*}R_{\mathbf{XX}}\left(t_{i},t_{j}\right)\geq0.</math> | + | <math class="inline">\sum_{i=1}^{n}\sum_{j=1}^{n}a_{i}a_{j}^{*}R_{\mathbf{XX}}\left(t_{i},t_{j}\right)\geq0.</math> |
Definition. White noise process | Definition. White noise process | ||
− | A random process <math>\mathbf{W}\left(t\right)</math> is a white noise process if <math>C_{\mathbf{WW}}\left(t_{1},t_{2}\right)=0 , \forall t_{1}\neq t_{2}</math> . | + | A random process <math class="inline">\mathbf{W}\left(t\right)</math> is a white noise process if <math class="inline">C_{\mathbf{WW}}\left(t_{1},t_{2}\right)=0 , \forall t_{1}\neq t_{2}</math> . |
Note | Note | ||
− | We will see that all non-trivial white noise processes have <math>C_{\mathbf{WW}}\left(t_{1},t_{2}\right)=q\left(t_{1}\right)\cdot\delta\left(t_{1}-t_{2}\right)</math> . | + | We will see that all non-trivial white noise processes have <math class="inline">C_{\mathbf{WW}}\left(t_{1},t_{2}\right)=q\left(t_{1}\right)\cdot\delta\left(t_{1}-t_{2}\right)</math> . |
Definition. Gaussian random process | Definition. Gaussian random process | ||
− | A random process <math>\mathbf{X}\left(t\right)</math> is called Gaussian random process if random variables <math>\mathbf{X}\left(t_{1}\right),\mathbf{X}\left(t_{2}\right),\cdots,\mathbf{X}\left(t_{n}\right)</math> are jointly Gaussian for any <math>n\in\mathbf{N}</math> and any set of sampling times <math>t_{1},t_{2},\cdots,t_{n}</math> . | + | A random process <math class="inline">\mathbf{X}\left(t\right)</math> is called Gaussian random process if random variables <math class="inline">\mathbf{X}\left(t_{1}\right),\mathbf{X}\left(t_{2}\right),\cdots,\mathbf{X}\left(t_{n}\right)</math> are jointly Gaussian for any <math class="inline">n\in\mathbf{N}</math> and any set of sampling times <math class="inline">t_{1},t_{2},\cdots,t_{n}</math> . |
Note | Note | ||
− | The n-th order characteristic function of a Gaussian random process is <math>\Phi_{\mathbf{X}\left(t_{1}\right)\cdots\mathbf{X}\left(t_{n}\right)}\left(\omega_{1},\cdots,\omega_{n}\right)=\exp\left\{ i\sum_{k=1}^{n}\eta_{\mathbf{X}}\left(t_{k}\right)\omega_{k}-\frac{1}{2}\sum_{j=1}^{n}\sum_{k=1}^{n}C_{\mathbf{XX}}\left(t_{j},t_{k}\right)\omega_{j}\omega_{k}\right\}</math> . | + | The n-th order characteristic function of a Gaussian random process is <math class="inline">\Phi_{\mathbf{X}\left(t_{1}\right)\cdots\mathbf{X}\left(t_{n}\right)}\left(\omega_{1},\cdots,\omega_{n}\right)=\exp\left\{ i\sum_{k=1}^{n}\eta_{\mathbf{X}}\left(t_{k}\right)\omega_{k}-\frac{1}{2}\sum_{j=1}^{n}\sum_{k=1}^{n}C_{\mathbf{XX}}\left(t_{j},t_{k}\right)\omega_{j}\omega_{k}\right\}</math> . |
Important Fact | Important Fact | ||
− | A Gaussian random process is completely characterized by <math>\eta_{\mathbf{X}}\left(t\right)=E\left[\mathbf{X}\left(t\right)\right]</math> and <math>C_{\mathbf{XX}}\left(t_{1},t_{2}\right)=E\left[\mathbf{X}\left(t_{1}\right)\mathbf{X}^{*}\left(t_{2}\right)\right]</math> . | + | A Gaussian random process is completely characterized by <math class="inline">\eta_{\mathbf{X}}\left(t\right)=E\left[\mathbf{X}\left(t\right)\right]</math> and <math class="inline">C_{\mathbf{XX}}\left(t_{1},t_{2}\right)=E\left[\mathbf{X}\left(t_{1}\right)\mathbf{X}^{*}\left(t_{2}\right)\right]</math> . |
3.1.3 Stationary Processes | 3.1.3 Stationary Processes | ||
Line 169: | Line 169: | ||
Definition. SSS | Definition. SSS | ||
− | A random process <math>\mathbf{X}\left(t\right)</math> is called stationary or strict-sense stationary (SSS) if its probalistic description is invariant to shifts in the time origin: | + | A random process <math class="inline">\mathbf{X}\left(t\right)</math> is called stationary or strict-sense stationary (SSS) if its probalistic description is invariant to shifts in the time origin: |
− | <math>F_{\mathbf{X}\left(t_{1}\right),\cdots,\mathbf{X}\left(t_{n}\right)}\left(x_{1},\cdots,x_{n}\right)=F_{\mathbf{X}\left(t_{1}+c\right),\cdots,\mathbf{X}\left(t_{n}+c\right)}\left(x_{1},\cdots,x_{n}\right)</math> | + | <math class="inline">F_{\mathbf{X}\left(t_{1}\right),\cdots,\mathbf{X}\left(t_{n}\right)}\left(x_{1},\cdots,x_{n}\right)=F_{\mathbf{X}\left(t_{1}+c\right),\cdots,\mathbf{X}\left(t_{n}+c\right)}\left(x_{1},\cdots,x_{n}\right)</math> |
− | for all <math>c\in\mathbf{R}</math> , for all <math>n\in\mathbf{N}</math> , and for all <math>t_{1},\cdots,t_{n}</math> . | + | for all <math class="inline">c\in\mathbf{R}</math> , for all <math class="inline">n\in\mathbf{N}</math> , and for all <math class="inline">t_{1},\cdots,t_{n}</math> . |
Definition. Jointly SSS | Definition. Jointly SSS | ||
Line 179: | Line 179: | ||
Two random processes are jointly SSS if their joint probabilistic description is invariant to shifts in the time origin: | Two random processes are jointly SSS if their joint probabilistic description is invariant to shifts in the time origin: | ||
− | <math>F_{\mathbf{X}\left(t_{1}\right),\cdots,\mathbf{X}\left(t_{n}\right),\mathbf{Y}\left(t_{1}\right),\cdots,\mathbf{Y}\left(t_{n}\right)}\left(x_{1},\cdots,x_{n},y_{1},\cdots,y_{n}\right)=F_{\mathbf{X}\left(t_{1}+c\right),\cdots,\mathbf{X}\left(t_{n}+c\right),\mathbf{Y}\left(t_{1}+c\right),\cdots,\mathbf{Y}\left(t_{n}+c\right)}\left(x_{1},\cdots,x_{n},y_{1},\cdots,y_{n}\right).</math> | + | <math class="inline">F_{\mathbf{X}\left(t_{1}\right),\cdots,\mathbf{X}\left(t_{n}\right),\mathbf{Y}\left(t_{1}\right),\cdots,\mathbf{Y}\left(t_{n}\right)}\left(x_{1},\cdots,x_{n},y_{1},\cdots,y_{n}\right)=F_{\mathbf{X}\left(t_{1}+c\right),\cdots,\mathbf{X}\left(t_{n}+c\right),\mathbf{Y}\left(t_{1}+c\right),\cdots,\mathbf{Y}\left(t_{n}+c\right)}\left(x_{1},\cdots,x_{n},y_{1},\cdots,y_{n}\right).</math> |
− | for all <math>c\in\mathbf{R}</math> , for all <math>n\in\mathbf{N}</math> , and for all <math>t_{1},\cdots,t_{n}</math> . | + | for all <math class="inline">c\in\mathbf{R}</math> , for all <math class="inline">n\in\mathbf{N}</math> , and for all <math class="inline">t_{1},\cdots,t_{n}</math> . |
Definition | Definition | ||
− | A complex random process <math>\mathbf{Z}\left(t\right)=\mathbf{X}\left(t\right)+i\mathbf{Y}\left(t\right)</math> where <math>\mathbf{X}\left(t\right)</math> and <math>\mathbf{Y}\left(t\right)</math> are real processes is SSS if <math>\mathbf{X}\left(t\right)</math> and <math>\mathbf{Y}\left(t\right)</math> are jointly SSS. | + | A complex random process <math class="inline">\mathbf{Z}\left(t\right)=\mathbf{X}\left(t\right)+i\mathbf{Y}\left(t\right)</math> where <math class="inline">\mathbf{X}\left(t\right)</math> and <math class="inline">\mathbf{Y}\left(t\right)</math> are real processes is SSS if <math class="inline">\mathbf{X}\left(t\right)</math> and <math class="inline">\mathbf{Y}\left(t\right)</math> are jointly SSS. |
Notes on stationary random processes | Notes on stationary random processes | ||
− | 1. The first-order (n=1 ) cdf or pdf of a stationary random process is independent of time: <math>f_{\mathbf{X}\left(t\right)}\left(x\right)=f_{\mathbf{X}\left(t+c\right)}\left(x\right),\quad\forall t\in\mathbf{R}.</math> | + | 1. The first-order (n=1 ) cdf or pdf of a stationary random process is independent of time: <math class="inline">f_{\mathbf{X}\left(t\right)}\left(x\right)=f_{\mathbf{X}\left(t+c\right)}\left(x\right),\quad\forall t\in\mathbf{R}.</math> |
− | 2. The second order (n=2 ) cdf or pdf is a function of time only through the time difference <math>\tau_{12}=t_{1}-t_{2} :</math> <math>f_{\mathbf{X}\left(t_{1}\right)\mathbf{X}\left(t_{2}\right)}\left(x_{1},x_{2}\right)=f_{\mathbf{X}\left(t_{1}+c\right)\mathbf{X}\left(t_{2}+c\right)}\left(x_{1},x_{2}\right)=f\left(x_{1},x_{2},\tau_{12}\right).</math> | + | 2. The second order (n=2 ) cdf or pdf is a function of time only through the time difference <math class="inline">\tau_{12}=t_{1}-t_{2} :</math> <math class="inline">f_{\mathbf{X}\left(t_{1}\right)\mathbf{X}\left(t_{2}\right)}\left(x_{1},x_{2}\right)=f_{\mathbf{X}\left(t_{1}+c\right)\mathbf{X}\left(t_{2}+c\right)}\left(x_{1},x_{2}\right)=f\left(x_{1},x_{2},\tau_{12}\right).</math> |
Definition. WSS | Definition. WSS | ||
− | • A random process <math>\mathbf{X}\left(t\right)</math> is called wide sense stationary (WSS) if it satisfies the following two conditions: | + | • A random process <math class="inline">\mathbf{X}\left(t\right)</math> is called wide sense stationary (WSS) if it satisfies the following two conditions: |
− | 1. <math>E\left[\mathbf{X}\left(t\right)\right]=\eta_{\mathbf{X}}\left(t\right)=\eta_{\mathbf{X}}\left(\mathbf{\text{constant}}\right)</math> . | + | 1. <math class="inline">E\left[\mathbf{X}\left(t\right)\right]=\eta_{\mathbf{X}}\left(t\right)=\eta_{\mathbf{X}}\left(\mathbf{\text{constant}}\right)</math> . |
− | 2. <math>E\left[\mathbf{X}\left(t_{1}\right)\mathbf{X}^{*}\left(t_{2}\right)\right]=R_{\mathbf{XX}}\left(t_{1},t_{2}\right)=R_{\mathbf{X}}\left(t_{1}-t_{2}\right)=R_{\mathbf{X}}\left(\tau\right)</math> where <math>\tau=t_{1}-t_{2}</math> . | + | 2. <math class="inline">E\left[\mathbf{X}\left(t_{1}\right)\mathbf{X}^{*}\left(t_{2}\right)\right]=R_{\mathbf{XX}}\left(t_{1},t_{2}\right)=R_{\mathbf{X}}\left(t_{1}-t_{2}\right)=R_{\mathbf{X}}\left(\tau\right)</math> where <math class="inline">\tau=t_{1}-t_{2}</math> . |
− | • For a WSS random process <math>\mathbf{X}\left(t\right)</math> , <math>E\left[\left|\mathbf{X}\left(t\right)\right|^{2}\right]=R_{\mathbf{XX}}\left(t_{1},t_{2}\right)|_{t_{1}=t,t_{2}=t}=R_{\mathbf{XX}}\left(t,t\right)=R_{\mathbf{X}}\left(t-t\right)=R_{\mathbf{X}}\left(0\right).</math> | + | • For a WSS random process <math class="inline">\mathbf{X}\left(t\right)</math> , <math class="inline">E\left[\left|\mathbf{X}\left(t\right)\right|^{2}\right]=R_{\mathbf{XX}}\left(t_{1},t_{2}\right)|_{t_{1}=t,t_{2}=t}=R_{\mathbf{XX}}\left(t,t\right)=R_{\mathbf{X}}\left(t-t\right)=R_{\mathbf{X}}\left(0\right).</math> |
− | • The autocovariance function of a WSS random process <math>\mathbf{X}\left(t\right)</math> is <math>C_{\mathbf{XX}}\left(t_{1},t_{2}\right)=E\left[\left(\mathbf{X}\left(t_{1}\right)-\eta_{\mathbf{X}}\right)\left(\mathbf{X}\left(t_{2}\right)-\eta_{\mathbf{X}}\right)^{*}\right]=R_{\mathbf{X}}\left(t_{1}-t_{2}\right)-\eta_{\mathbf{X}}\eta_{\mathbf{X}}^{*}=R_{\mathbf{X}}\left(\tau\right)-\eta_{\mathbf{X}}\eta_{\mathbf{X}}^{*}=C_{\mathbf{X}}\left(\tau\right)</math> where <math>\tau=t_{1}-t_{2}</math> . | + | • The autocovariance function of a WSS random process <math class="inline">\mathbf{X}\left(t\right)</math> is <math class="inline">C_{\mathbf{XX}}\left(t_{1},t_{2}\right)=E\left[\left(\mathbf{X}\left(t_{1}\right)-\eta_{\mathbf{X}}\right)\left(\mathbf{X}\left(t_{2}\right)-\eta_{\mathbf{X}}\right)^{*}\right]=R_{\mathbf{X}}\left(t_{1}-t_{2}\right)-\eta_{\mathbf{X}}\eta_{\mathbf{X}}^{*}=R_{\mathbf{X}}\left(\tau\right)-\eta_{\mathbf{X}}\eta_{\mathbf{X}}^{*}=C_{\mathbf{X}}\left(\tau\right)</math> where <math class="inline">\tau=t_{1}-t_{2}</math> . |
Note | Note | ||
− | If <math>\mathbf{X}\left(t\right)</math> is a SSS random process, then it is WSS. The converse is NOT true. The Gaussian random process is an exception. | + | If <math class="inline">\mathbf{X}\left(t\right)</math> is a SSS random process, then it is WSS. The converse is NOT true. The Gaussian random process is an exception. |
Theorem | Theorem | ||
− | If <math>\mathbf{X}\left(t\right)</math> is a WSS random process and it is Gaussian, then it is SSS. | + | If <math class="inline">\mathbf{X}\left(t\right)</math> is a WSS random process and it is Gaussian, then it is SSS. |
Proof | Proof | ||
− | Assume <math>\mathbf{X}\left(t\right)</math> has mean <math>\eta_{\mathbf{X}}\left(t\right)=E\left[\mathbf{X}\left(t\right)\right]=\eta_{\mathbf{X}}</math> and covariance <math>C_{\mathbf{XX}}\left(t_{1},t_{2}\right)=C_{\mathbf{X}}\left(t_{1}-t_{2}\right)</math> . Then the random variables <math>\mathbf{X}\left(t_{1}+c\right),\cdots,\mathbf{X}\left(t_{n}+c\right)</math> have joint characteristic function. | + | Assume <math class="inline">\mathbf{X}\left(t\right)</math> has mean <math class="inline">\eta_{\mathbf{X}}\left(t\right)=E\left[\mathbf{X}\left(t\right)\right]=\eta_{\mathbf{X}}</math> and covariance <math class="inline">C_{\mathbf{XX}}\left(t_{1},t_{2}\right)=C_{\mathbf{X}}\left(t_{1}-t_{2}\right)</math> . Then the random variables <math class="inline">\mathbf{X}\left(t_{1}+c\right),\cdots,\mathbf{X}\left(t_{n}+c\right)</math> have joint characteristic function. |
− | <math>\Phi_{\mathbf{X}\left(t_{1}+c\right)\cdots\mathbf{X}\left(t_{n}+c\right)}\left(\omega_{1},\cdots,\omega_{n}\right)=\exp\left\{ i\sum_{k=1}^{n}\eta_{\mathbf{X}}\left(t_{k}+c\right)\omega_{k}-\frac{1}{2}\sum_{j=1}^{n}\sum_{k=1}^{n}C_{\mathbf{XX}}\left(t_{j}+c,t_{k}+c\right)\omega_{j}\omega_{k}\right\}</math><math>=\exp\left\{ i\sum_{k=1}^{n}\eta_{\mathbf{X}}\left(t_{k}\right)\omega_{k}-\frac{1}{2}\sum_{j=1}^{n}\sum_{k=1}^{n}C_{\mathbf{XX}}\left(t_{j},t_{k}\right)\omega_{j}\omega_{k}\right\} </math><math>=\Phi_{\mathbf{X}\left(t_{1}\right)\cdots\mathbf{X}\left(t_{n}\right)}\left(\omega_{1},\cdots,\omega_{n}\right).</math> | + | <math class="inline">\Phi_{\mathbf{X}\left(t_{1}+c\right)\cdots\mathbf{X}\left(t_{n}+c\right)}\left(\omega_{1},\cdots,\omega_{n}\right)=\exp\left\{ i\sum_{k=1}^{n}\eta_{\mathbf{X}}\left(t_{k}+c\right)\omega_{k}-\frac{1}{2}\sum_{j=1}^{n}\sum_{k=1}^{n}C_{\mathbf{XX}}\left(t_{j}+c,t_{k}+c\right)\omega_{j}\omega_{k}\right\}</math><math class="inline">=\exp\left\{ i\sum_{k=1}^{n}\eta_{\mathbf{X}}\left(t_{k}\right)\omega_{k}-\frac{1}{2}\sum_{j=1}^{n}\sum_{k=1}^{n}C_{\mathbf{XX}}\left(t_{j},t_{k}\right)\omega_{j}\omega_{k}\right\} </math><math class="inline">=\Phi_{\mathbf{X}\left(t_{1}\right)\cdots\mathbf{X}\left(t_{n}\right)}\left(\omega_{1},\cdots,\omega_{n}\right).</math> |
− | <math>\therefore\mathbf{X}\left(t\right)</math> is SSS. | + | <math class="inline">\therefore\mathbf{X}\left(t\right)</math> is SSS. |
---- | ---- |
Latest revision as of 11:47, 30 November 2010
3.1 Definitions
Stochastic process
The idea of a stochastic process is a straightforward extension of that of random variable. Instead of mapping each $ \omega\in\mathcal{S} $ of a random experiment to a number $ \mathbf{X}\left(\omega\right) $ , we map it to a function of time $ \mathbf{X}\left(t,\omega\right) $ that is called sample function.
Note
There is nothing random about the sample functions. The randomness comes from the underlying random experiment.
Note
If we pick a particular point in time $ t=t_{1} $ , we have $ \mathbf{X}\left(t_{1},\omega\right) $ that is a random variable.
Definition
A stochastic process or random process defined on $ \left(\mathcal{S},\mathcal{F},\mathcal{P}\right) $ is a famility of random variables
$ \left\{ \mathbf{X}\left(t\right):t\in\mathbf{T}\right\} $
indexed by $ t $ , where the index set $ \mathbf{T} $ can be discrete or continuous.
Note
1. If $ \mathbf{T} $ is an uncountable subset of $ \mathbf{R} $ , $ \mathbf{X}\left(t\right) $ is called a continuous-time random process.
2. If $ \mathbf{T} $ is an discrete set, $ \mathbf{X}\left(t\right) $ is called a discrete-time random process.
3. $ \mathbf{X}\left(t\right) $ is called a discrete-state random process if for all $ t\in\mathbf{T} $ , it takes on values from a discrete set. Otherwise it is called a continuous-state random process.
Note
Thermal noise is continuous-time continuous state random process.
Notation
We will use the notation $ \mathbf{X}\left(t\right) $ to represent a random process (just as we use $ \mathbf{X} $ to represent a random variable). Technically we should write $ \mathbf{X}\left(t,\omega\right) $ or $ \mathbf{X}\left(\cdot,\cdot\right) $ (just as technically we should write $ \mathbf{X}\left(\omega\right) $ or $ \mathbf{X}\left(\cdot\right) $ for a random variable).
$ \mathbf{X}\left(\cdot,\cdot\right):\mathbf{T}\times\mathcal{S}\longrightarrow\mathbf{R}. $
note
• $ \mathbf{X}\left(t,\omega\right) $ or $ \mathbf{X}\left(\cdot,\cdot\right) $ is a random process.
• $ \mathbf{X}\left(t_{0},\omega\right) $ or $ \mathbf{X}\left(t_{0},\cdot\right) $ is a random variable for fixed </math>t_{0}</math> .
• $ \mathbf{X}\left(t,\omega_{0}\right) $ or $ \mathbf{X}\left(\cdot,\omega_{0}\right) $ for a fixed $ \omega_{0}\in\mathcal{S} $ is a function of time or sample function.
• $ \mathbf{X}\left(t_{0},\omega_{0}\right) $ is a real number.
3.1.1 Statistics of Stochastic Processes
First-order CDF and PDF
We use CDFs and PDFs to characterize the probalistic behavior of a random process. A random process sampled at a point in time is a random variable. We can consider its CDF or PDF.
Definition
The first-order CDF of a random process $ \mathbf{X}\left(t\right) $ is
$ F_{\mathbf{X}\left(t\right)}\left(x\right)\triangleq P\left(\left\{ X\left(t\right)\leq x\right\} \right)=F_{\mathbf{X}\left(t\right)}\left(x,t\right) $.
The first-order PDF of a random process $ \mathbf{X}\left(t\right) $ is
$ f_{\mathbf{X}\left(t\right)}\left(x\right)=\frac{dF_{\mathbf{X}\left(t\right)}\left(x\right)}{dx}=f_{\mathbf{X}\left(t\right)}\left(x,t\right). $
The n-th order CDF and PDF
We are often interested in the joint behavior of $ \mathbf{X}\left(t\right) $ sample at more than one point in time. First-order description is not sufficient.
Definition
The n-th order CDF of a random process $ \mathbf{X}\left(t\right) $ is
$ F_{\mathbf{X}\left(t_{1}\right),\cdots,\mathbf{X}\left(t_{n}\right)}\left(x_{1},\cdots,x_{n}\right) $
and the n-th order PDF is
$ f_{\mathbf{X}\left(t_{1}\right),\cdots,\mathbf{X}\left(t_{n}\right)}\left(x_{1},\cdots,x_{n}\right)=\frac{\partial^{n}F_{\mathbf{X}\left(t_{1}\right),\cdots,\mathbf{X}\left(t_{n}\right)}\left(x_{1},\cdots,x_{n}\right)}{\partial x_{1}\cdots\partial x_{n}}. $
Of particular interest are the second order $ \left(n=2\right) $ CDFs and PDFs, $ F_{\mathbf{X}\left(t_{1}\right),\mathbf{X}\left(t_{2}\right)}\left(x_{1},x_{2};t_{1},t_{2}\right) $ and $ f_{\mathbf{X}\left(t_{1}\right),\mathbf{X}\left(t_{2}\right)}\left(x_{1},x_{2};t_{1},t_{2}\right) $ .
3.1.2 General Properties
Fact
A complex random process $ \mathbf{Z}\left(t\right)=\mathbf{X}\left(t\right)+i\mathbf{Y}\left(t\right) $ where $ \mathbf{X}\left(t\right) $ and $ \mathbf{Y}\left(t\right) $ are real jointly-distributed random processes, is completely characterized by
$ F_{\mathbf{X}\left(t_{1}\right),\cdots,\mathbf{X}\left(t_{n}\right),\mathbf{Y}\left(t_{1}\right),\cdots,\mathbf{Y}\left(t_{n}\right)}\left(x_{1},\cdots,x_{n},y_{1},\cdots,y_{n},t_{1},\cdots,t_{n}\right) $
for all $ n\in\mathbf{N} $ and all $ t_{1},\cdots,t_{n} $ . (This is a very hard result to proof.)
Fact. The real version
A real random process $ \mathbf{X}\left(t\right) $ is completely chracterized by the joint CDF
$ F_{\mathbf{X}\left(t_{1}\right),\cdots,\mathbf{X}\left(t_{n}\right)}\left(x_{1},\cdots,x_{n};t_{1},\cdots,t_{n}\right) $
for all $ n\in\mathbf{N} $ and all $ t_{1},\cdots,t_{n} $ .
Defintion. Autocorrelation function
For a random process $ \mathbf{X}\left(t\right) $ , real or complex, the autocorrelation function of $ \mathbf{X}\left(t\right) $ is
$ R_{\mathbf{XX}}\left(t_{1},t_{2}\right)\triangleq E\left[\mathbf{X}\left(t_{1}\right)\mathbf{X}^{*}\left(t_{2}\right)\right]. $
Note
$ R_{\mathbf{XX}}\left(t_{1},t_{2}\right) $
Note
For $ \mathbf{X}\left(t\right) $ real
$ R_{\mathbf{XX}}\left(t_{1},t_{2}\right)=E\left[\mathbf{X}\left(t_{1}\right)\mathbf{X}\left(t_{2}\right)\right] $
and
$ R_{\mathbf{XX}}\left(t_{1},t_{2}\right)=R_{\mathbf{XX}}\left(t_{2},t_{1}\right). $
Note
$ E\left[\left|\mathbf{X}\left(t\right)\right|^{2}\right]=E\left[\mathbf{X}\left(t\right)\mathbf{X}^{*}\left(t\right)\right]=R_{\mathbf{XX}}\left(t,t\right). $
$ E\left[\left|\mathbf{X}\left(t\right)\right|^{2}\right]\geq0\Longrightarrow R_{\mathbf{XX}}\left(t,t\right)\geq0,\quad\forall t. $
Definition. Mean
The mean of a random process $ \mathbf{X}\left(t\right) $ is
$ \eta_{\mathbf{X}}\left(t\right)=E\left[\mathbf{X}\left(t\right)\right]. $
If $ \mathbf{X}\left(t\right) $ is a complex random process with $ \mathbf{X}\left(t\right)=\mathbf{X}_{R}\left(t\right)+i\mathbf{X}_{I}\left(t\right) $ , then $ E\left[\mathbf{X}\left(t\right)\right]=E\left[\mathbf{X}\left(t\right)=\mathbf{X}_{R}\left(t\right)+i\mathbf{X}_{I}\left(t\right)\right]=E\left[\mathbf{X}_{R}\left(t\right)\right]+iE\left[\mathbf{X}_{I}\left(t\right)\right]. $
Definition. Autocovariance function
The autocovariance function of a random process $ \mathbf{X}\left(t\right) $ with mean $ \eta_{\mathbf{X}}\left(t\right)=E\left[\mathbf{X}\left(t\right)\right] $ is
$ C_{\mathbf{XX}}\left(t_{1},t_{2}\right) $
Fact
The autocorrelation function $ R_{\mathbf{XX}}\left(t_{1},t_{2}\right) $ is a non-negative definite function: for any set of numbers $ \left\{ a_{i}\right\} $ and times $ \left\{ t_{i}\right\} $ , and any $ n\in\mathbf{N} $
$ \sum_{i=1}^{n}\sum_{j=1}^{n}a_{i}a_{j}^{*}R_{\mathbf{XX}}\left(t_{i},t_{j}\right)\geq0. $
Definition. White noise process
A random process $ \mathbf{W}\left(t\right) $ is a white noise process if $ C_{\mathbf{WW}}\left(t_{1},t_{2}\right)=0 , \forall t_{1}\neq t_{2} $ .
Note
We will see that all non-trivial white noise processes have $ C_{\mathbf{WW}}\left(t_{1},t_{2}\right)=q\left(t_{1}\right)\cdot\delta\left(t_{1}-t_{2}\right) $ .
Definition. Gaussian random process
A random process $ \mathbf{X}\left(t\right) $ is called Gaussian random process if random variables $ \mathbf{X}\left(t_{1}\right),\mathbf{X}\left(t_{2}\right),\cdots,\mathbf{X}\left(t_{n}\right) $ are jointly Gaussian for any $ n\in\mathbf{N} $ and any set of sampling times $ t_{1},t_{2},\cdots,t_{n} $ .
Note
The n-th order characteristic function of a Gaussian random process is $ \Phi_{\mathbf{X}\left(t_{1}\right)\cdots\mathbf{X}\left(t_{n}\right)}\left(\omega_{1},\cdots,\omega_{n}\right)=\exp\left\{ i\sum_{k=1}^{n}\eta_{\mathbf{X}}\left(t_{k}\right)\omega_{k}-\frac{1}{2}\sum_{j=1}^{n}\sum_{k=1}^{n}C_{\mathbf{XX}}\left(t_{j},t_{k}\right)\omega_{j}\omega_{k}\right\} $ .
Important Fact
A Gaussian random process is completely characterized by $ \eta_{\mathbf{X}}\left(t\right)=E\left[\mathbf{X}\left(t\right)\right] $ and $ C_{\mathbf{XX}}\left(t_{1},t_{2}\right)=E\left[\mathbf{X}\left(t_{1}\right)\mathbf{X}^{*}\left(t_{2}\right)\right] $ .
3.1.3 Stationary Processes
Definition. SSS
A random process $ \mathbf{X}\left(t\right) $ is called stationary or strict-sense stationary (SSS) if its probalistic description is invariant to shifts in the time origin:
$ F_{\mathbf{X}\left(t_{1}\right),\cdots,\mathbf{X}\left(t_{n}\right)}\left(x_{1},\cdots,x_{n}\right)=F_{\mathbf{X}\left(t_{1}+c\right),\cdots,\mathbf{X}\left(t_{n}+c\right)}\left(x_{1},\cdots,x_{n}\right) $
for all $ c\in\mathbf{R} $ , for all $ n\in\mathbf{N} $ , and for all $ t_{1},\cdots,t_{n} $ .
Definition. Jointly SSS
Two random processes are jointly SSS if their joint probabilistic description is invariant to shifts in the time origin:
$ F_{\mathbf{X}\left(t_{1}\right),\cdots,\mathbf{X}\left(t_{n}\right),\mathbf{Y}\left(t_{1}\right),\cdots,\mathbf{Y}\left(t_{n}\right)}\left(x_{1},\cdots,x_{n},y_{1},\cdots,y_{n}\right)=F_{\mathbf{X}\left(t_{1}+c\right),\cdots,\mathbf{X}\left(t_{n}+c\right),\mathbf{Y}\left(t_{1}+c\right),\cdots,\mathbf{Y}\left(t_{n}+c\right)}\left(x_{1},\cdots,x_{n},y_{1},\cdots,y_{n}\right). $
for all $ c\in\mathbf{R} $ , for all $ n\in\mathbf{N} $ , and for all $ t_{1},\cdots,t_{n} $ .
Definition
A complex random process $ \mathbf{Z}\left(t\right)=\mathbf{X}\left(t\right)+i\mathbf{Y}\left(t\right) $ where $ \mathbf{X}\left(t\right) $ and $ \mathbf{Y}\left(t\right) $ are real processes is SSS if $ \mathbf{X}\left(t\right) $ and $ \mathbf{Y}\left(t\right) $ are jointly SSS.
Notes on stationary random processes
1. The first-order (n=1 ) cdf or pdf of a stationary random process is independent of time: $ f_{\mathbf{X}\left(t\right)}\left(x\right)=f_{\mathbf{X}\left(t+c\right)}\left(x\right),\quad\forall t\in\mathbf{R}. $
2. The second order (n=2 ) cdf or pdf is a function of time only through the time difference $ \tau_{12}=t_{1}-t_{2} : $ $ f_{\mathbf{X}\left(t_{1}\right)\mathbf{X}\left(t_{2}\right)}\left(x_{1},x_{2}\right)=f_{\mathbf{X}\left(t_{1}+c\right)\mathbf{X}\left(t_{2}+c\right)}\left(x_{1},x_{2}\right)=f\left(x_{1},x_{2},\tau_{12}\right). $
Definition. WSS
• A random process $ \mathbf{X}\left(t\right) $ is called wide sense stationary (WSS) if it satisfies the following two conditions:
1. $ E\left[\mathbf{X}\left(t\right)\right]=\eta_{\mathbf{X}}\left(t\right)=\eta_{\mathbf{X}}\left(\mathbf{\text{constant}}\right) $ .
2. $ E\left[\mathbf{X}\left(t_{1}\right)\mathbf{X}^{*}\left(t_{2}\right)\right]=R_{\mathbf{XX}}\left(t_{1},t_{2}\right)=R_{\mathbf{X}}\left(t_{1}-t_{2}\right)=R_{\mathbf{X}}\left(\tau\right) $ where $ \tau=t_{1}-t_{2} $ .
• For a WSS random process $ \mathbf{X}\left(t\right) $ , $ E\left[\left|\mathbf{X}\left(t\right)\right|^{2}\right]=R_{\mathbf{XX}}\left(t_{1},t_{2}\right)|_{t_{1}=t,t_{2}=t}=R_{\mathbf{XX}}\left(t,t\right)=R_{\mathbf{X}}\left(t-t\right)=R_{\mathbf{X}}\left(0\right). $
• The autocovariance function of a WSS random process $ \mathbf{X}\left(t\right) $ is $ C_{\mathbf{XX}}\left(t_{1},t_{2}\right)=E\left[\left(\mathbf{X}\left(t_{1}\right)-\eta_{\mathbf{X}}\right)\left(\mathbf{X}\left(t_{2}\right)-\eta_{\mathbf{X}}\right)^{*}\right]=R_{\mathbf{X}}\left(t_{1}-t_{2}\right)-\eta_{\mathbf{X}}\eta_{\mathbf{X}}^{*}=R_{\mathbf{X}}\left(\tau\right)-\eta_{\mathbf{X}}\eta_{\mathbf{X}}^{*}=C_{\mathbf{X}}\left(\tau\right) $ where $ \tau=t_{1}-t_{2} $ .
Note
If $ \mathbf{X}\left(t\right) $ is a SSS random process, then it is WSS. The converse is NOT true. The Gaussian random process is an exception.
Theorem
If $ \mathbf{X}\left(t\right) $ is a WSS random process and it is Gaussian, then it is SSS.
Proof
Assume $ \mathbf{X}\left(t\right) $ has mean $ \eta_{\mathbf{X}}\left(t\right)=E\left[\mathbf{X}\left(t\right)\right]=\eta_{\mathbf{X}} $ and covariance $ C_{\mathbf{XX}}\left(t_{1},t_{2}\right)=C_{\mathbf{X}}\left(t_{1}-t_{2}\right) $ . Then the random variables $ \mathbf{X}\left(t_{1}+c\right),\cdots,\mathbf{X}\left(t_{n}+c\right) $ have joint characteristic function.
$ \Phi_{\mathbf{X}\left(t_{1}+c\right)\cdots\mathbf{X}\left(t_{n}+c\right)}\left(\omega_{1},\cdots,\omega_{n}\right)=\exp\left\{ i\sum_{k=1}^{n}\eta_{\mathbf{X}}\left(t_{k}+c\right)\omega_{k}-\frac{1}{2}\sum_{j=1}^{n}\sum_{k=1}^{n}C_{\mathbf{XX}}\left(t_{j}+c,t_{k}+c\right)\omega_{j}\omega_{k}\right\} $$ =\exp\left\{ i\sum_{k=1}^{n}\eta_{\mathbf{X}}\left(t_{k}\right)\omega_{k}-\frac{1}{2}\sum_{j=1}^{n}\sum_{k=1}^{n}C_{\mathbf{XX}}\left(t_{j},t_{k}\right)\omega_{j}\omega_{k}\right\} $$ =\Phi_{\mathbf{X}\left(t_{1}\right)\cdots\mathbf{X}\left(t_{n}\right)}\left(\omega_{1},\cdots,\omega_{n}\right). $
$ \therefore\mathbf{X}\left(t\right) $ is SSS.