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[communication principle] Chapter 2 -- deterministic signal

2022-07-26 11:42:00 【Bald baby】

Know the signal ： Its reference is a definite and predictable signal at any time , Usually, it can be expressed by mathematical formula as its value at any time
- According to whether it has periodic repeatability , It can be divided into periodic signals and aperiodic signals
- Distinguish according to whether the energy is limited , It can be divided into energy signal and power signal
use S Represent the current or voltage of the signal to calculate the signal power , If the value of signal voltage and current changes with time , be S It can be rewritten as time t Function of S(t), here , Signal energy E It should be the integral of the instantaneous power of the signal , among E The unit of is Joule J
$\int_{-∞}^{∞}{s^2(t)dt}$
If the signal of energy is a positive finite value , This signal is called energy signal , The average power of the signal is defined as
$\lim_{T\to∞}\frac{1}{T}\int_{-\frac{T}{2}}^{\frac{T}{2}}s^2(t)dt$

Energy signal ： Its energy is equal to a finite positive value , But the average power is zero
Power signal ： Its average power is equal to a finite positive value , But the energy is infinite

Be careful ： The classification of energy signals and power signals is also used for unascertained signals

Frequency characteristics , Expressed by the distribution of each frequency component
Frequency characteristic of signal
1. Spectrum of power signal
2. Spectral density of energy signal
3. Energy spectral density of energy signal
4. Power spectral density of power signal

Set a periodic power signal s(t) Both periods are T₀, Then its spectrum function is defined as
$C_n = C(nf_0) = \frac{1}{T}\int_{-\frac{T_0}{2}}^{\frac{T_0}{2}}{s(t)e^{-j2πnf_0t}dt}$
The periodic signal can be expanded into the following Fourier series
$\sum_{n = -∞}^{∞}{C_ne^{\frac{j2πnt}{T_0}}}$
The Dirichlet condition that can mathematically expand periodic functions into Fourier series , General signals can be satisfied
n = 0 when ,
$C_0 = \frac{1}{T_0}\int_{-\frac{T_0}{2}}^{\frac{T_0}{2}}{s(t)dt}$
He is a signal s(t) Time average of , The DC component

Set an energy signal as s(t), Then its Fourier transform S(f) Defined as its spectral density
$\int_{-∞}^{∞}{s(t)e^{-j2πft}dt}$
and S(f) The inverse Fourier transform of is the original signal
$\int_{-∞}^{∞}S(f)e^{j2πft}df$

Important conclusions

For gate functions ( rectangular )
- S(f) = area ·sinc( Variable · Width )
- s(t) = area ·sinc( Variable · Width )
Trigonometrically shaped function
- S(f) = area ·sinc²( Variable · Width /2)
- s(t) = area ·sinc²( Variable · Width /2)

Set an energy signal s(t) The energy of is E, Then the energy of this signal is
$\int_{-∞}^{∞}{s^2(t)}dt$
If the Fourier transform of this signal ( Spectral density ) by S(f), Then we can get from Barcelona theorem
$\int_{-∞}^{∞}{s^2(t)}dt = \int_{-∞}^{∞}{|S(f)|^2}df$

Because the power signal has infinite energy , So first, signal s(t) Truncate to a length equal to T A truncated signal of s_T(t), The truncated signal becomes an energy signal
For this energy signal , Fourier transform can be used to calculate its energy spectral density |S_T(f)|²
According to Barcelona theorem
$\int_{-\frac{T}{2}}^{\frac{T}{2}}{s_T^2(t)}dt = \int_{-∞}^{∞}{|S_T(f)|^2}df$

$\lim_{T\to∞}{\frac{1}{T}}{|S_T(f)|^2}$

The power spectral density of the signal P(f)
$\lim_{T\to∞}{\frac{1}{T}}{|S_T(f)|^2}$
The signal power is
$\lim_{T\to∞}{\frac{1}{T}}\int_{-∞}^{∞}{|S_T(f)|^2}df = \int_{-∞}^{∞}{P(f)}df$

Energy signal s(t) The autocorrelation function of is defined as
$\int_{-∞}^{∞}{s(t)s(t+τ)}dt -∞ < τ < ∞$
The autocorrelation function reflects a signal and delay τ The degree of correlation between the same signals after , Autocorrelation function R(τ) And time t irrelevant , Only time difference τ of , When τ = 0 when , Autocorrelation function of energy signal R(0) Equal to the energy of the signal , namely
$\int_{-∞}^{∞}{s^2(t)}dt = E$
among ,E Is the energy of the energy signal
R(τ) yes τ Even function of ,R(τ) = R(-τ)

Power signal s(t) The autocorrelation function of is defined as
$\lim_{T\to∞}\frac{1}{T}\int_{-\frac{T}{2}}^{\frac{T}{2}}{s(t)s(t+τ)}dt -∞ < τ < ∞$
When τ = 0 when , Autocorrelation function of power signal R(0) Equal to the average power of the signal , namely
$\lim_{T\to∞}\frac{1}{T}\int_{-\frac{T}{2}}^{\frac{T}{2}}{s^2(t)}dt = P$
among ,P Is the power of the signal
For periodic power signals , The definition of autocorrelation function can be rewritten as
$\frac{1}{T_0}\int_{-\frac{T_0}{2}}^{\frac{T_0}{2}}{s(t)s(t+τ)}dt -∞ < τ < ∞$
Autocorrelation function of periodic power signal R(τ) And its power spectral density P(f) The relationship between them is Fourier transform , namely P(f) The inverse Fourier transform of is R(τ), and R(τ) The Fourier transform of is the power spectral density , namely
$\int_{-∞}^{∞}{R(τ)e^{-j2πfτ}}dτ$

Two energy signals s₁(t) and s₂(t) The cross-correlation function of is defined as
$R_{12}(τ) = \int_{-∞}^{∞}{s_{1}(t)s_{2}(t+τ)}dt -∞ < τ < ∞$
Cross correlation function and time t irrelevant , Only time difference τ of , The cross-correlation function is related to the order in which two signals are multiplied
$R_{12}(τ) = R_{21}(-τ)$

Two power signals s₁(t) and s₂(t) The cross-correlation function of is defined as
$R_{12}(τ) = \lim_{T\to∞}\frac{1}{T}\int_{-∞}^{∞}{s_{1}(t)s_{2}(t+τ)}dt -∞ < τ < ∞$
Cross correlation function and time t irrelevant , Only time difference τ of , The cross-correlation function is related to the order in which two signals are multiplied
$R_{12}(τ) = R_{21}(-τ)$
If the period of two periodic power signals is the same , Then the definition of its cross-correlation function can be written as
$R_{12}(τ) = \frac{1}{T}\int_{-\frac{T}{2}}^{\frac{T}{2}}{s_{1}(t)s_{2}(t+τ)}dt -∞ < τ < ∞$