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radar:pulsecompression [2018/06/08 08:34] – [Reduction of the lateral lobes] romagnoliradar:pulsecompression [2026/04/28 18:24] (current) – mauro
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- For the generic signal >ok! make attention to the titles size!  --- //[[webmaster@localhost|DokuWiki Administrator]] 2018/04/23 12:56// 
  
->  Where do the figures come from? Please cite the document as decribed in [[:start|Welcome!]] --- //[[webmaster@localhost|DokuWiki Administrator]] 2018/05/03 16:16// 
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-> please use caption for tables and figures  ---  //[[webmaster@localhost|DokuWiki Administrator]] 2018/05/03 16:13// 
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-> please tables should be tables not pictures   ---  //[[webmaster@localhost|DokuWiki Administrator]] 2018/05/03 16:13// 
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-> please use numbered equation (if they are not in-line with the text ---  //[[webmaster@localhost|DokuWiki Administrator]] 2018/05/03 16:13// 
  
 ====== Basic concepts concerning Matched filters ====== ====== Basic concepts concerning Matched filters ======
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    * Where $\theta$ is the constant filter time; $\theta = ( \pi B)^{-1} $ and $B$ is the bandwidth.    * Where $\theta$ is the constant filter time; $\theta = ( \pi B)^{-1} $ and $B$ is the bandwidth.
-Changing the value of $B$ an optimal condition can be found, in which in the case of the rectangular pulse is equal to: $B=\frac{0.4}{T}$. +Changing the value of $B$ an optimal condition can be found, in which in the case of the rectangular pulse is equal to: $B=\frac{0.4}{T}$, in the case of an ideal Pass-Band filter the best band is $B=\frac{1.4}{T}$. Cases of practical interest have optimum bands whose values fall between these two extremes. The optimum band has a value approximately equal to the length of the rectangular pulse.
-In the case of an ideal Pass-Band filter the best band $B=\frac{1.4}{T}$. Cases of practical interest have optimum bands whose values fall between these two extremes. The optimum band has a value approximately equal to the length of the rectangular pulse.+
 The procedure that allows the calculation of the impulsive response of the matched filter will be discussed below. The procedure that allows the calculation of the impulsive response of the matched filter will be discussed below.
 Suppose that the receiving chain of the radar (See the figure below) is characterized by an impulse response $h(t)$. Let $s(t)$ and $y(t)$ be respectively the input useful signals and an output of the chain. Suppose that the receiving chain of the radar (See the figure below) is characterized by an impulse response $h(t)$. Let $s(t)$ and $y(t)$ be respectively the input useful signals and an output of the chain.
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 \begin{equation} \left ( \frac{\tilde{S}}{N} \right)_{out} =  \frac{ \left | \int_{-inf}^{+inf}S(f)H(f)e^{j2\pi ft_{1}}\,df \right |^2 }{\int_{-inf}^{+inf} N(f) \left | H(f) \right |^{2}\,df }   \end{equation} \begin{equation} \left ( \frac{\tilde{S}}{N} \right)_{out} =  \frac{ \left | \int_{-inf}^{+inf}S(f)H(f)e^{j2\pi ft_{1}}\,df \right |^2 }{\int_{-inf}^{+inf} N(f) \left | H(f) \right |^{2}\,df }   \end{equation}
  
-The value of the ratio signal/noise $S/N$ can be obtained considering the effective value  (not the peak value) of the signal. Therefore we have:+The value of the ratio $S/N$ can be obtained considering the effective value  (not the peak value) of the signal. Therefore we have:
  
 \begin{equation} \left ( \frac{S}{N} \right)_{out} = \frac{1}{2} \left( \frac{\tilde{S}}{N} \right)_{out} \end{equation} \begin{equation} \left ( \frac{S}{N} \right)_{out} = \frac{1}{2} \left( \frac{\tilde{S}}{N} \right)_{out} \end{equation}
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-In the figure below //(a)//, the signal is shown, and in the figure //(b)// the relative impulsive response of the matched filter +In the figure below //(a)//, the signal is showed, and in the figure //(b)// the relative impulsive response of the matched filter  is showed.
  
  
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 </figure> </figure>
  
-The values of $R$ and $C$ are chosen to maximize the ratio $SNR$ at the output of the filter, it comes that for $ B=0.4/T, T $ is the duration of the pulse, The filter provides the maximum ratio $SNR$ at the output. The filter below is called //"Pseudo adapted ".//+The values of $R$ and $C$ are chosen to maximize the ratio $SNR$ at the output of the filter, it comes that for $ B=0.4/T, T $ is the duration of the pulse. The filter below is called //"Pseudo adapted ".//
  
  
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 With what has been said so far, in the continuous time domain, can be extended on the discrete-time case. Digital signal processing is becoming more and more common in every branch of communications and radar techniques, owing to the possibility of an exact and reliable implementation of complicated algorithms. With what has been said so far, in the continuous time domain, can be extended on the discrete-time case. Digital signal processing is becoming more and more common in every branch of communications and radar techniques, owing to the possibility of an exact and reliable implementation of complicated algorithms.
-So it is quite easy to implement a matched filter for every waveform, using the infinite impulse response //(IIR)// or finite impulse response //(FIR)// techniques for digital filters.[(cite:Advanced)] A remarkable exception is surface acoustic wave (SAW) devices, allowing straightforward and exact implementation at IF of the matched filter to any waveform of finite duration (the main limitation being the maximum duration of the waveform to be compressed, owing to the limited velocity of the acoustic wave, namely 3 mm/ jus, and the need to keep the dimension of the substrate acceptable), however we will discuss this topic later.+So it is quite easy to implement a matched filter for every waveform, using the infinite impulse response //(IIR)// or finite impulse response //(FIR)// techniques for digital filters.[(cite:Advanced)] A remarkable exception is surface acoustic wave (SAW) devices, allowing straightforward and exact implementation at IF of the matched filter to any waveform of finite duration (the main limitation being the maximum duration of the waveform to be compressed, owing to the limited velocity of the acoustic wave, and the need to keep the dimension of the substrate acceptable), however we will discuss this topic later.
  
 ====== Range accuracy and sampling problems ====== ====== Range accuracy and sampling problems ======
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-In absence of noice, the received signal would have the shape illustrated in the figure with a dashed line  **__c)__** .+In absence of noice, the received signal has the shape showed in the figure with a dashed line  **__c)__** .
 Because of the noise we have to modify the signal that could be as in the figure ** __c)__ **. Because of the noise we have to modify the signal that could be as in the figure ** __c)__ **.
 As we said previously, the radar is able to detect the target if the received signal exceeds the threshold $V_{T}$ at the instant in which the distance is measured, measuring the delay time $T_{R}$ with which the echo of the signal is received. As we said previously, the radar is able to detect the target if the received signal exceeds the threshold $V_{T}$ at the instant in which the distance is measured, measuring the delay time $T_{R}$ with which the echo of the signal is received.
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-Suppose that $t=T$ we have a maximum of the signal in the output. To find a temporal position $ (t=T) $ of the maximum, we can use a derivator. +Suppose that $t=T$ and we have a maximum of the signal in the output. To find a temporal position $ (t=T) $ of the maximum, we can use a derivator. 
-This procedure is the optimal linear estimator of the delay, therefore of distance. In the following is illustrated the procedure that gives the least average square deviation; it can be demonstrated that is equal to:+This procedure is the optimal linear estimator of the delay, therefore of distance. In the following is showed the procedure that gives the least average square deviation; it can be demonstrated that is equal to:
  
  
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-Where $ X\left( f \right ) $, is the energy spectrum of the transmitted base-band waveform ( spectrum of the complex envelope ).+$ X\left( f \right ) $ is the energy spectrum of the transmitted base-band waveform ( spectrum of the complex envelope ).
 Hence if we use a matched filter, we can also measure the distance of the target with a precision that depends on the band and the $SNR$. Hence if we use a matched filter, we can also measure the distance of the target with a precision that depends on the band and the $SNR$.
 ** High accuracy is necessary for a tracking radar which only processes the pulses associated with the target being tracked. **  ** High accuracy is necessary for a tracking radar which only processes the pulses associated with the target being tracked. ** 
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 The requirements a) and b) are conflicting. However, a method exists for improving the resolution that is The requirements a) and b) are conflicting. However, a method exists for improving the resolution that is
-based on the encoding of the signal transmitted by the radar: // ** The pulse compression** //+based on the encoding of the signal transmitted by the radar: // ** The pulse compression** //.
  
  
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 \begin{equation} h \left ( t \right ) = k cos \left ( 2 \pi f_{0} t - \frac{ \mu t^{2} }{2}  \right ) \: \: \: \:   for   \: \: \: 0<t<T \end{equation} \begin{equation} h \left ( t \right ) = k cos \left ( 2 \pi f_{0} t - \frac{ \mu t^{2} }{2}  \right ) \: \: \: \:   for   \: \: \: 0<t<T \end{equation}
  
-In the figure below there is the impulsive response of the matched filter of the // Chirp signal //+In the figure below there is the impulsive response of the matched filter of the // Chirp signal //.
  
  
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 <figure label> <figure label>
 {{ :media:block_diagram_to_reduce_lateral_lobes.jpg?500 |}} {{ :media:block_diagram_to_reduce_lateral_lobes.jpg?500 |}}
-<caption> block diagram of the system to reduce the lateral lobes [(cite:Teoria)]</caption>+<caption> Block diagram of the system to reduce the lateral lobes [(cite:Teoria)]</caption>
 </figure> </figure>
  
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 The term $ W(f) $ introduces a mismatch in reception. An unmatched amplitude response results in a The term $ W(f) $ introduces a mismatch in reception. An unmatched amplitude response results in a
-degradation of the "signal-to-noise" ratio, although frequency weighing is a good method of reducing the side lobes. Among the most important and most widely used weighing functions, $ W(f) $ are the distribution of //Dolph-Chebyshev// and //Taylor's distribution//. Dolph-Chebyshev's weighing is not physically feasible because it requires infinite gain at the extremes of the band. At the design stage, the distribution of Dolph-Chebyshev is used as a comparison term with other achievable methods that try to approximate it, such as Taylor, Hamming, etc. as chirp. +degradation of the "signal-to-noise" ratio, although frequency weighing is a good method of reducing the side lobes. Among the most important and most widely used weighing functions, $ W(f) $ are the distribution of //Dolph-Chebyshev// and //Taylor's distribution//. Dolph-Chebyshev's weighing is not physically feasible because it requires infinite gain at the extremes of the band. At the design stage, the distribution of Dolph-Chebyshev is used as a comparison term with other achievable methods that try to approximate it, such as Taylor, Hamming, etc.. 
 It is possible to see in the image below about of Taylor and Dolph-Chebyshev's methods. It is possible to see in the image below about of Taylor and Dolph-Chebyshev's methods.
  
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-Usually, the matched filter to a coded sequence can be made in base band or intermediate frequency. In the modern systems the first solution is adopted, using phase and quadrature **I** and **Q** samples. The transmitted signal is represented by the convolution of a rectangular pulse duration $ \tau $ with the sequence of $N$ components **I** and **Q**  describing the code. After the sampling operation there is a sequence of phase samples of the received waveform :+Usually, the matched filter to a coded sequence can be made in base band or intermediate frequency. In the modern systems the first solution is adopted. The transmitted signal is represented by the convolution of a rectangular pulse duration $ \tau $ with the sequence of $N$ components **I** and **Q**  describing the code. After the sampling operation there is a sequence of phase samples of the received waveform :
  
 \begin{equation} \begin{equation}
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 //Application of Digital/Analog pulse compression and final considerations with advantages and disadvantages of pulse compression.// //Application of Digital/Analog pulse compression and final considerations with advantages and disadvantages of pulse compression.//
  
-As already mentioned in the previous paragraphs; A pulse compression radar, transmits a coded signal with "//low//" peak power and "//long//" time duration. The transmitted signal normally has a rectangular envelope of //T// seconds duration coded in phase or frequency. When received, the signal is "//compressed//" by the matched filter or by filtering with a certain degree of mismatch; The compressed signal lasts $\tau$ seconds, with $ \tau = T / C $ being,  //C// a larger, and sometimes much larger than the unit, called the "**compression ratio**". To obtain the desired duration of $\tau$, the signal must occupy a bandwidth approximately equal to $1 / \tau$. The compressed signal has, in addition to the main peak width $ \tau $, lateral lobes that can be reduced by suitable filters that cause the above-mentioned disadvantage, resulting in an enlargement of the primary lobe and a loss in the peak signal-to-noise ratio.+As already mentioned in the previous paragraphs; A pulse compression radar, transmits a coded signal with "//low//" peak power and "//long//" time duration. The transmitted signal normally has a rectangular envelope of //T// seconds duration coded in phase or frequency. When received, the signal is "//compressed//" by the matched filter or by filtering with a certain degree of mismatch; The compressed signal lasts $\tau$ seconds, with $ \tau = T / C $, being  //C// a larger, and sometimes much larger than the unit, called the "**compression ratio**". To obtain the desired duration of $\tau$, the signal must occupy a bandwidth approximately equal to $1 / \tau$. The compressed signal has, in addition to the main peak width $ \tau $, lateral lobes that can be reduced by suitable filters that cause the above-mentioned disadvantage, resulting in an enlargement of the primary lobe and a loss in the peak signal-to-noise ratio.
  
 ====Limitations of pulse compression==== ====Limitations of pulse compression====
  
-Pulse compression has some disadvantages. It requires a transmitter that can be readily modulated and a receiver with a matched filter more sophisticated than that of a conventional pulse radar. Although it may be more complex than a conventional long-pulse radar. The equipment for high-power pulse compression radar is more practical than that one required by a short-pulse radar with the same pulse energy. When limiting is employed, there can be small-target suppression and possibly spurious false-targets as well [( cite:RadarSystems)] A pulse compression radar has also the following advantages over a radar with the same coverage (same ratio $2E / N_{0} $ with same average power):+Pulse compression has some disadvantages. It requires a transmitter that can be readily modulated and a receiver with a matched filter more sophisticated than that of a conventional pulse radar. Although it may be more complex than a conventional long-pulse radar. The equipment for high-power pulse compression radar is more practical than that one required by a short-pulse radar with the same pulse energy. When limiting is employed, there can be small-target suppression and possibly spurious false-targets as well [( cite:RadarSystems)]. A pulse compression radar has also the following advantages over a radar with the same coverage (same ratio $2E / N_{0} $ with same average power):
  
  
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