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radar:pulsecompression [2018/06/06 22:11] – romagnoliradar:pulsecompression [2026/04/28 18:24] (current) – mauro
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->  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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 ====== Basic concepts concerning Matched filters ====== ====== Basic concepts concerning Matched filters ======
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 Furthermore, was illustrated the detection criterion commonly adopted in radar systems, based on the comparison of the measured voltage $v(t)$, with an opportune threshold value $V_{T}$. Furthermore, was illustrated the detection criterion commonly adopted in radar systems, based on the comparison of the measured voltage $v(t)$, with an opportune threshold value $V_{T}$.
 The choice of the threshold value could be carried out according to the criteria described in the previous paragraph. The choice of the threshold value could be carried out according to the criteria described in the previous paragraph.
-Before introducing new concepts, referring to the radar equation, the $SNR_{min}$ depends only from the $P_{d}$ and of the $P_{fa}$ assigned, ( we remember that $P_{d}$ is //the probability of detection of the target //  and $P_{fa}$ is //the probability of false alarm // ) in this case, $SNR_{min}$ should be characterized. Starting from the definition of signal/noise, and from the expression of process power of noise at the end of the linear system, $\sigma^{2} = k_{b}T_{0}B_{n}GF$, ensures that the minimum power of the useful signal is: +Before introducing new concepts, referring to the radar equation, the $SNR_{min}$ depends only on the $P_{d}$ and on the $P_{fa}$ assigned, ( we remember that $P_{d}$ is //the probability of detection of the target //  and $P_{fa}$ is //the probability of false alarm // ) in this case, $SNR_{min}$ should be characterized. Starting from the definition of signal/noise, and from the expression of process power of noise at the end of the linear system, $\sigma^{2} = k_{b}T_{0}B_{n}GF$, ensures that the minimum power of the useful signal is: 
-$S{min}= SNR_{min}Fk_{b}T_{0}B_{n}$. The revelation is carried out after the receiver, and as the equivalent noise band $B_{n} $varies, the power of signals and noise also vary, it is necessary to introduce the concept of //matched filter//, namely is a type of filter $H(f)$ that maximizes the $SNR$ ratio.+$S{min}= SNR_{min}Fk_{b}T_{0}B_{n}$, where the quantities are referred 
 + at the input. The revelation is carried out after the receiver, and as the equivalent noise band $B_{n} $varies, the power of signals and noise also vary, it is necessary to introduce the concept of //matched filter//, namely is a type of filter $H(f)$ that maximizes the $SNR$ ratio.
  
 To better understand the definition of the matched filter in reception, it is a good rule to start with some qualitative concepts. To better understand the definition of the matched filter in reception, it is a good rule to start with some qualitative concepts.
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    * Suppose you have a receiving chain that has a transfer function with a certain bandwidth.    * Suppose you have a receiving chain that has a transfer function with a certain bandwidth.
  
-If the signal (sum of useful signal and the noise) has the shape indicated in figure (a), at the receiver's output, is expected to have a signal + noise like in figure (b).+If the signal (sum of useful signal and the noise) has the shape showed in figure (a), the receiver's output is expected to have a signal + noise like in figure (b).
  
  
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 If there was a filter with a higher band, in addition to a greater share of the useful power, a greater noise power would have been taken. It is intuitive to understand that what has been said can't be advantageous. If there was a filter with a higher band, in addition to a greater share of the useful power, a greater noise power would have been taken. It is intuitive to understand that what has been said can't be advantageous.
-However, we expect a filter to exist with optimum bandwidth, that maximizes the output signal-to-noise ratio in a precise instant of time.+However, we expect that a filter with optimum bandwidth exists, that maximizes the output signal-to-noise ratio in a precise instant of time.
  
 To better understand, suppose we have received a rectangular signal as shown below To better understand, suppose we have received a rectangular signal as shown below
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 It is easy to notice that in the figure above, the ratio $SNR$ depends from the instant of observation. It is easy to notice that in the figure above, the ratio $SNR$ depends from the instant of observation.
 In particular at the instance $t = t_{A}$, when the circuit $RC$ is fully loaded, we expected that the ratio $SNR$ is maximum, it is appropriate to identify the moment in which the ratio $SNR$ is maximum because the detection of the target occurs with a comparison of a threshold. In particular at the instance $t = t_{A}$, when the circuit $RC$ is fully loaded, we expected that the ratio $SNR$ is maximum, it is appropriate to identify the moment in which the ratio $SNR$ is maximum because the detection of the target occurs with a comparison of a threshold.
-Formalizing, suppose we can have a filter with a $B$ variable band, in the figure below, is illustrated the trends of the useful signal $S$, and the noise $N$, when the band $B$  of the filter varies.+Formalizing, suppose we can have a filter with a $B$ variable band, the figure below shows, the trends of the useful signal $S$, and the noise $N$, when the band $B$  of the filter varies.
  
  
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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.
-If you change the value of $B$ we found a condition of maximum, 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 chain receiving of the radar (See the figure below) is characterized by an impulse response $h(t)$. Are $s(t)$ and $y(t)$ 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. 
-Suppose that the instant time $t_{1}$ in which the signal $y(t)$ are the maximum noted value.+Suppose that the instant time $t_{1}$ in which the signal $y(t)$ is the maximum noted value.
  
  
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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 previous calculation, we assumed that $s(t)$ is an energy signal and that $E$ is the energy of the signal. In the previous calculation, we assumed that $s(t)$ is an energy signal and that $E$ is the energy of the signal.
-It was therefore shown that the set ratios of $SNR$, are superiorly limited by the value $2E/N_{0}$, in the case that the process is white. +It was therefore shown that the set ratios of $SNR$, are superiorly limited by the value $2E/N_{0}$, in the case of a white process. 
  
 If we find the transfer function such as: If we find the transfer function such as:
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-In the figure below //(a)//, the signal is illustrated, 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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 [(cite:Advanced>> title     : Advanced Radar Techniques and systems  authors   : Gaspare Galati [(cite:Advanced>> title     : Advanced Radar Techniques and systems  authors   : Gaspare Galati
 publisher : Peter Peregrinus published : 1993 pages     : 51 )] publisher : Peter Peregrinus published : 1993 pages     : 51 )]
-Shortly, with respect to telecommunications systems in the strict sense, the radar is characterized by the fact that the waveform of the echo of the target is known. In fact, the point target, resends a signal whose waveform is attenuated and phase shifted by the transmitted signal.+With respect to telecommunications systems in the strict sense, the radar is characterized by the fact that the waveform of the echo of the target is known. In fact, the point target, resends a signal whose waveform is attenuated and phase shifted by the transmitted signal.
 The purpose of the radar is to identify the presence and the characteristics of the target, not to decode the received signal. The purpose of the radar is to identify the presence and the characteristics of the target, not to decode the received signal.
  
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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 results 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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 </figure> </figure>
  
-The maximum ratio of $SNR$ is equal to 0.8 times the optimum. This is due to the fact that the filter is (slightly) misfit. In reality, it is possible to choose different filters, like ( Chebyshev filter, Bessel filter,ecc..). +The maximum $SNR$ is equal to 0.8 times the optimum. This is due to the fact that the filter is (slightly) misfit. In reality, it is possible to choose different filters, like ( Chebyshev filter, Bessel filter,ecc..). 
-To choose the type of filters, it is necessary to calculate the loss of the filters, respect to the optimal ideal case (see the table below).+To choose the type of filters, it is necessary to calculate the loss of the filters, with respect to the optimal ideal case (see the table below).
  
  
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 =====Discrete-time optimum filtering===== =====Discrete-time optimum filtering=====
  
-With what has been said so far, in the continuous time domain, it is possible to extend all the considerations of the previous section to 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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-If we don't have noise, 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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-//Basic concepts regard sampling losses, some examples and introduction to the pulse compression//+//Basic concepts regarding sampling losses, some examples and introduction to the pulse compression//
  
 In the surveillance radars, the precision is limited not only to the presence of noise but also to the fact that the received signal will be sampled. The sampling step is related to the band of the transmitted signal, below is illustrated the degradation effect, produced by sampling. Suppose a signal in output from the matched filter in baseband (without noise). Suppose we uniformly sample the output signal with a step equal to the duration of the transmitted pulse $T$ and that the first sample is taken at the instant **A** which is a random value. As a result, referring to the figure below, there will only be two samples associated with the received signal. Suppose that only the first of the two exceeds the threshold value. This leads to assert that the target is at a temporal distance that falls between // 0 and T //. In the surveillance radars, the precision is limited not only to the presence of noise but also to the fact that the received signal will be sampled. The sampling step is related to the band of the transmitted signal, below is illustrated the degradation effect, produced by sampling. Suppose a signal in output from the matched filter in baseband (without noise). Suppose we uniformly sample the output signal with a step equal to the duration of the transmitted pulse $T$ and that the first sample is taken at the instant **A** which is a random value. As a result, referring to the figure below, there will only be two samples associated with the received signal. Suppose that only the first of the two exceeds the threshold value. This leads to assert that the target is at a temporal distance that falls between // 0 and T //.
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 Referring to the figure above, if both samples exceed the decision threshold, it might be thought, that the maximum of the output filter ( and therefore the temporal position of the target ) will be close to the sample that is higher. If, however, several samples exceed the threshold voltage, it can be expected that there will be more than one target. However, it is not possible to derive this information by operating on the single pulse received: it is also necessary to operate in azimuth and if it is possible, also in Doppler. Referring to the figure above, if both samples exceed the decision threshold, it might be thought, that the maximum of the output filter ( and therefore the temporal position of the target ) will be close to the sample that is higher. If, however, several samples exceed the threshold voltage, it can be expected that there will be more than one target. However, it is not possible to derive this information by operating on the single pulse received: it is also necessary to operate in azimuth and if it is possible, also in Doppler.
-Customarily the radar resolution is defined in probabilistic terms for the reasons illustrated below. +Customarily the radar resolution is defined in probabilistic terms for the reasons showed below. 
-Suppose that $ n = 1 $, and the position of the first sample, is in a weak area. If we increase the sampling frequency, for example,// n= 2,3,4...//, it reduces the probability of not detecting the point where the signal has the maximum, sees the figure below.+Suppose that $ n = 1 $, and that the position of the first sample, is in a weak area. If we increase the sampling frequency, for example,// n= 2,3,4...//, it is reduced the probability of not detecting the point where the signal has the maximum, sees the figure below.
  
  
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 From the figure above, we can observe that for $ n = 1 $ there is a loss of $1.5dB $ or $ 2 dB$, while for $ n = 2$ there is a loss only for $0.3dB$ or $0.4 dB$. The value of  $n$ can be split up, which means that for example; if $n =1.5 $, there are three samples in a time equal to the duration of two pulses. From the figure above, we can observe that for $ n = 1 $ there is a loss of $1.5dB $ or $ 2 dB$, while for $ n = 2$ there is a loss only for $0.3dB$ or $0.4 dB$. The value of  $n$ can be split up, which means that for example; if $n =1.5 $, there are three samples in a time equal to the duration of two pulses.
  
-Assuming that two pulses have been received at two distinct targets, at the output of the matched filter, the situation is illustrated in the figure below.+Assuming that two pulses have been received at two distinct targets, at the output of the matched filter, the situation is showed in the figure below.
  
  
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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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 The Pulse compression involves the transmission of a long coded pulse and the processing of the received echo to obtain a relatively narrow pulse in case of pulse radar. The increased detection capability of a long-pulse radar system is achieved while retaining the range resolution capability of a narrow-pulse system. Several advantages are obtained, the transmission of long pulses permits a more efficient use of the average power capability of the radar, the generation of high peak power signals is avoided. The average power of the radar may be increased without increasing the pulse repetition frequency (PRF) and, hence, decreasing the radar's unambiguous range. An increased system resolving capability in doppler is also obtained as a result of the use of the long pulse. In addition, the radar is less vulnerable to interfering signals that differ from the coded transmitted signal. The Pulse compression involves the transmission of a long coded pulse and the processing of the received echo to obtain a relatively narrow pulse in case of pulse radar. The increased detection capability of a long-pulse radar system is achieved while retaining the range resolution capability of a narrow-pulse system. Several advantages are obtained, the transmission of long pulses permits a more efficient use of the average power capability of the radar, the generation of high peak power signals is avoided. The average power of the radar may be increased without increasing the pulse repetition frequency (PRF) and, hence, decreasing the radar's unambiguous range. An increased system resolving capability in doppler is also obtained as a result of the use of the long pulse. In addition, the radar is less vulnerable to interfering signals that differ from the coded transmitted signal.
-A long pulse may be generated from a narrow pulse. A narrow pulse contains a large number of frequency components with a precise phase relationship between them. If the relative phases are changed by a phase-distorting filter, the frequency components combine to produce a stretched a pulse. This expanded pulse is the pulse that is transmitted. The received echo is processed in the receiver by a compression filter. The compression filter readjusts the relative phases of the frequency components so that a narrow or compressed pulse is again produced. The pulse compression ratio is the ratio of the width of the expanded pulse to that of the compressed pulse.+A long pulse may be generated from a narrow pulse. A narrow pulse contains a large number of frequency components with a precise phase relationship between them. If the relative phases are changed by a phase-distorting filter, the frequency components combine to produce a stretched a pulse. This expanded pulse is the pulse that is transmitted. The received echo is processed in the receiver by a compression filter. The compression filter readjusts the relative phases of the frequency components so that a narrow or compressed pulse is again produced. The pulse compression ratio is the ratio of the width of the expanded pulse to that and the compressed pulse.
 //A pulse compression radar is a practical implementation of a matched-filter system//. The output of the matched-filter section is the compressed pulse, which is given by the inverse Fourier transform of the product of the signal spectrum [(cite:HandBook)]. //A pulse compression radar is a practical implementation of a matched-filter system//. The output of the matched-filter section is the compressed pulse, which is given by the inverse Fourier transform of the product of the signal spectrum [(cite:HandBook)].
  
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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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-The frequency response of the matched filter, for high values of the "BT compression" product (for $BT> 1$), tends to a rectangle with a base //B// and centre frequency $f_{0}$, with which it takes on the appearance as the figure below. +The frequency response of the matched filter, for high values of the "BT compression" product (for $BT> 1$), tends to a rectangle with a base //B// and central frequency $f_{0}$, as the figure below.
  
  
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-The //Pulse compression// allows improving the resolution in distance. In particular pulse compression allows a radar to utilize a long pulse to achieve large radiated energy, but simultaneously to obtain the range resolution of a short pulse. +The //Pulse compression// allows an improvement in the resolution in distance. In particular pulse compression allows a radar to utilize a long pulse to achieve large radiated energy, but simultaneously to obtain the range resolution of a short pulse. 
-We indicate with **$r$** the minimum distance between two objects that the radar is able to discriminate, as we know, $ r = ct/2 $, where $c$ is the light speed, and $t$ is the duration of the pulse. In the case of the compressed pulse, we have $t= \tau$, while in the case where there a compression of the pulse, $t=T$. Being  $t<<T$, there is a considerable improvement, // the disadvantage is that the compressed pulse, has lateral lobes until $13.26 dB$ under the maximum; This value take the name of ** PSLR ** (**Peak sidelobe Ratio**) //+We indicate with **$r$** the minimum distance between two objects that the radar is able to discriminate, as we know, $ r = ct/2 $, where $c$ is the light speed, and $t$ is the duration of the pulse. In the case of the compressed pulse, we have $t= \tau$, while in the case where there a compression of the pulse, $t=T$. Being  $t<<T$, there is a considerable improvement, // the disadvantage is that the compressed pulse, has lateral lobes up to $13.26 dB$ below the maximum; This value takes the name of ** PSLR ** (**Peak sidelobe Ratio**) //
  
  
-When there are many objects, that is, there are multiple targets whose radial distances are less than $ c/T2 $ where  $T$ is the duration of the transmitted pulse, due to the high lobes, the interference could arise that would obscure the weakest signals. The effect is illustrated in the figure below. Techniques used to reduce these lobes: signal weighting in transmission or in reception or non-linear frequency modulation. +When there are many objects, that is, there are multiple targets whose radial distances are less than $ c/T2 $ where  $T$ is the duration of the transmitted pulse, due to the high lobes, the interference could cancel the weakest signals. The effect is illustrated in the figure below. Techniques used to reduce these lobes: signal weighting in transmission or in reception or non-linear frequency modulation. 
  
  
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 </figure> </figure>
  
-In reality, imperfections in matched filtering could be considered as differences in the amplitude of the matched filter or in the phase of the matched filter, compared with the ideal. In the figure below are illustrated a chirp signal after a pulse compression, in particular, we can see the central signal that is surrounded by sidelobes, the main lobes being at $13.5 dB$ below the central signal[(cite:Advanced)].+In reality, imperfections in matched filtering could be considered as differences in the amplitude of the matched filter or in the phase of the matched filter, compared with the ideal. The figure below shows a chirp signal after a pulse compression, in particular, we can see the central signal that is surrounded by sidelobes, the main lobes being at $13.5 dB$ below the central signal[(cite:Advanced)].
 In the following paragraphs, will be discussed methods to reduce the lateral lobes. In the following paragraphs, will be discussed methods to reduce the lateral lobes.
  
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 ====Reduction of the lateral lobes==== ====Reduction of the lateral lobes====
  
-As previously mentioned, when using pulse compression, the problem of the high level of the side lobes occurs. The chirp signal becomes a short signal surrounded by parasitic sidelobes, the two closest sidelobes having a level of $13.3 dB$ below the main signal. This means that, if a parasitic target is very close to a useful one with an equivalent echoing area more than $13 dB$ above the useful one, it will mask it that is generally not acceptable[(cite:Advanced)].+As previously mentioned, when using pulse compression, the problem of the high level of the side lobes occurs. The chirp signal becomes a short signal surrounded by sidelobes, the two closest sidelobes having a level of $13.3 dB$ below the main signal. This means that, if a parasitic target is very close to a useful one with an equivalent echoing area more than $13 dB$ above the useful one, it will mask it that is generally not acceptable[(cite:Advanced)].
 However, there are some techniques for reducing the level of the lateral lobes, namely:  However, there are some techniques for reducing the level of the lateral lobes, namely: 
  
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-In high-power radar applications, it is preferred to transmit a constant power, so the technic ** a) **  is not advisable, as it would cause a loss of power. Instead, the other two techniques are more suitable, the technique ** b) ** allows to reduce the lateral lobes at the expense of an enlarged compressed pulse. The technique ** c) **, has the disadvantage of being more complex to realize and being sensitive to doppler displacements.+In high-power radar applications, it is preferred to transmit a constant power, so the technique ** a) **  is not advisable, as it would cause a loss of power. Instead, the other two techniques are more suitable, the technique ** b) ** allows to reduce the lateral lobes at the expense of an enlarged compressed pulse. The technique ** c) **, has the disadvantage of being more complex to realize and being sensitive to doppler displacements.
 A method for obtaining the desired waveform (at low lateral lobes) with frequency weighting is shown in the figure below. A method for obtaining the desired waveform (at low lateral lobes) with frequency weighting is shown in the figure below.
  
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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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 \begin{equation} W(f) = \int_{-inf}^{+inf} g(t)exp(-j2 \pi ft) \ ,dt \end{equation} \begin{equation} W(f) = \int_{-inf}^{+inf} g(t)exp(-j2 \pi ft) \ ,dt \end{equation}
  
-However, the result is not generally zero for, as required for a limited bandwidth such as chirp. 
 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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-In the figure below, is illustrated the Hamming function with, band  ** $ B = 1Mhz $ **. +The figure below shows the Hamming function with, band  ** $ B = 1Mhz $ **. 
  
 <figure label> <figure label>
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-If we have a target at $ 3Km $ of distance, it will reply after twenty $\mu s$, but as the radar is still transmitting, the response cannot be received. Then, in conclusion, signals associated with targets whose distance is less than:+If we have a target at $ 3Km $ of distance, it will reply after 20 $\mu s$, but as the radar is still transmitting, the response cannot be received. Then, in conclusion, signals associated with targets whose distance is less than:
  
 \begin{equation} R_{0} = \frac{cT}{2} \end{equation} \begin{equation} R_{0} = \frac{cT}{2} \end{equation}
  
-They can not be seen by the radar, $T$ is the duration of the transmitted pulse. To be able to see nearby targets, it is common to transmit on a different carrier frequency and after the "long" pulse of long duration $T$, a short-term pulse $\Delta T $ unencoded (see the figure below).+They cannot be seen by the radar, $T$ is the duration of the transmitted pulse. To be able to see nearby targets, it is common to transmit on a different carrier frequency and after the "long" pulse of long duration $T$, a short-term pulse $\Delta T $ unencoded (see the figure below).
  
 <figure label> <figure label>
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 //Introduction of PSK (Phase Shift Keying), application of Barker code, some concepts concerning the Doppler Effects mismatching and the effect of pulse compression on clutter.// //Introduction of PSK (Phase Shift Keying), application of Barker code, some concepts concerning the Doppler Effects mismatching and the effect of pulse compression on clutter.//
  
-A // Phase-coded pulse compression//, in this form of pulse compression a long pulse of duration $T$ is divided into $N$ subpulses each of width $\tau$. The phase of each subpulse is chosen to be either 0 or $\pi$ radians. If the selection of the 0, $\pi$ phase is made at random, the waveform approximates a noise-modulated signal with a thumbtack ambiguity function. The binary choice of 0 or $\pi$ phase for each subpulse may be made random. However, some random selections may be better suited than others for radar application. One criterion for the selection of a good "random" phase-coded waveform is that the output of the matched filter is the autocorrelation of the input signal for which it is matched. [(cite:RadarSystems>> Title     : Introduction to radar Systems   : Merril I. Skolnik Publisher : McGrow-Hill Published : 1981 Pages     : 429)]+A // Phase-coded pulse compression//, in this form of pulse compression a long pulse of duration $T$ is divided into $N$ subpulses each of width $\tau$. The phase of each subpulse is chosen to be either 0 or $\pi$ radiants. If the selection of the 0, $\pi$ phase is made at random, the waveform approximates a noise-modulated signal with a thumbtack ambiguity function. The binary choice of 0 or $\pi$ phase for each subpulse may be made random. However, some random selections may be better suited than others for radar application. One criterion for the selection of a good "random" phase-coded waveform is that the output of the matched filter is the autocorrelation of the input signal for which it is matched. [(cite:RadarSystems>> Title     : Introduction to radar Systems   : Merril I. Skolnik Publisher : McGrow-Hill Published : 1981 Pages     : 429)]
    
  
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-The band associated with the signal is equal to: $1/ \tau $ and consequently there is a compression ratio $BT$ equal to $BT = 5 $. In the figure below, is shown the envelope of the signal ** (a) ** and the relative function of autocorrelation, that is the output of the matched filter.+The band associated to the signal is equal to: $1/ \tau $ and consequently there is a compression ratio $BT$ equal to $BT = 5 $. In the figure below, is shown the envelope of the signal ** (a) ** and the relative function of autocorrelation, that is the output of the matched filter.
  
 <figure label> <figure label>
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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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 </table> </table>
  
-The use of Barker codes and more convenient of BSPK, because the to Barker sequences have real coefficients equal to //+1 or  -1//. The binary coding of the radar signal phases is only one of the possible choices. You can also choose to assume as a sequence of phases the one associated with the chirp signal. This means that a quadratic phase shift is produced by passing from a duration sub-pulse of duration $ \tau$ to the next. In this case, is obtained the polyphase code of the type **discretized chirp**. +The use of Barker codes and more convenient than BSPK, because the to Barker sequences have real coefficients equal to //+1 or  -1//. The binary coding of the radar signal phases is only one of the possible choices. You can also choose to assume as a sequence of phases the one associated with the chirp signal. This means that a quadratic phase shift is produced by passing from a duration sub-pulse of duration $ \tau$ to the next. In this case, is obtained the polyphase code of the type **discretized chirp**. 
  
    
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-The signal being transmitted is a discrete chirp signal with a sampling step that respects Shannon's theorem conditions. Consequently, we assume that $ \tau = 1/B $, from the discretized chirp, derive some of the codes described below.+The signal being transmitted is a discrete chirp signal with a sampling step that respects Shannon's theorem conditions.
 The Chirp and Barker encodings are not the only possible ones. There are other types of encodings that can be The Chirp and Barker encodings are not the only possible ones. There are other types of encodings that can be
 used in the radar system: used in the radar system:
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 ====Doppler Effects Mismatching==== ====Doppler Effects Mismatching====
  
-If the target is fixed, the back-diffused signal is a replica of the transmitted signal. If the target is moving the signal is changed and the receiving filter, in general, is not matched to the received signal. In practice, it is difficult to build a matched filter for each shift doppler that may occur. The effects of an adaptation for doppler effect should, therefore, be evaluated case by case. Suppose you have transmitted a waveform whose analytical signal is given by+If the target is fixed, the back-scattered signal is a replica of the transmitted signal. If the target is moving the signal is changed and the receiving filter, in general, is not matched to the received signal. In practice, it is difficult to build a matched filter for each shift doppler that may occur. The effects of an adaptation for doppler effect should, therefore, be evaluated case by case. Suppose you have transmitted a waveform whose analytical signal is given by
  
 \begin{equation} \begin{equation}
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 If the transmitted pulse is encoded as in figure **a)**, in the case of a doppler effect it is expected that at the instant //**A**// , the phase assumes a value equal to $\phi$ while at the distance //**B**//  it is even at $ \phi +\pi \: \: etc.$. If, on the other hand, there is a doppler effect at time A, there is a phase value equal to $ \phi + \Delta \phi $ while in B, we have $ \phi + 2\Delta \phi + \pi $ etc. If the transmitted pulse is encoded as in figure **a)**, in the case of a doppler effect it is expected that at the instant //**A**// , the phase assumes a value equal to $\phi$ while at the distance //**B**//  it is even at $ \phi +\pi \: \: etc.$. If, on the other hand, there is a doppler effect at time A, there is a phase value equal to $ \phi + \Delta \phi $ while in B, we have $ \phi + 2\Delta \phi + \pi $ etc.
-The magnitude of the variation allows understanding if it is possible to use or not, for example, the codes of Barker. In fact if $\tau $ is small, then $\Delta _{\phi}$ is negligible. If the Received signal has a long duration, at the same interval the sampling $\tau$, the shift doppler $f_{D}$, and the term $k \Delta \phi$ can be harmful because it can alter the sequence of phases, at the limit it can also transforms a phase 0 with $\pi$ and vice-versa.+The magnitude of the variation allows to understand if it is possible to use or not, for example, the codes of Barker. In fact if $\tau $ is small, then $\Delta _{\phi}$ is negligible. If the Received signal has a long duration, at the same interval the sampling $\tau$, the shift doppler $f_{D}$, and the term $k \Delta \phi$ can be harmful because it can alter the sequence of phases, at the limit it can also transforms a phase 0 with $\pi$ and vice-versa.
  
  
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 If there are a very large number of pieces of clutter in the resolution cell of the radar. The pulse compression, because it represents the division of the range resolution by the pulse compression ratio, represents the division of the number of pieces of clutter and then the power of the clutter by the same ratio. But this is a rare situation. In the most cases, the numbers of pieces of clutter are not sufficient to consider that the effect of pulse compression is only to reduce the power of the If there are a very large number of pieces of clutter in the resolution cell of the radar. The pulse compression, because it represents the division of the range resolution by the pulse compression ratio, represents the division of the number of pieces of clutter and then the power of the clutter by the same ratio. But this is a rare situation. In the most cases, the numbers of pieces of clutter are not sufficient to consider that the effect of pulse compression is only to reduce the power of the
 clutter by the pulse compression ratio. clutter by the pulse compression ratio.
-In reality a system //without pulse compression//, doppler filtering of excellent quality will be required, while in the case of //pulse compression// for example; if the required detection probability is $0.9$, it will be necessary only to use a system which cancels the clutter as well as the useful targets mixed with it (Doppler filtering with a fairly poor 'sub-clutter visibility' will be convenient)[(cite:Advanced)].+In a system //without pulse compression//, doppler filtering of excellent quality will be required, while in the case of //pulse compression// for example; if the required detection probability is $0.9$, it will be necessary only to use a system which cancels the clutter as well as the useful targets mixed with it (Doppler filtering with a fairly poor 'sub-clutter visibility' will be convenient)[(cite:Advanced)].
  
  
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-From these properties **a)** and **b)**, putting //E = 1// (signal-energy normalization) results that if we try to lower the sidelobes  $| X (t, f_{d}) |$,(since the volume and maximum value of the ambiguity function are fixed), there will be an enlargement of the main lobe. It is not possible to obtain a "pointy graph", which would allow good discrimination as much as is desired in both radius and distance. The best you can try to get is an ambiguity function with a narrow peak and with sufficiently low side lobes, see the figure below.+From these properties **a)** and **b)**, putting //E = 1// (signal-energy normalization) comes that if we try to lower the sidelobes  $| X (t, f_{d}) |$,(since the volume and maximum value of the ambiguity function are fixed), there will be an enlargement of the main lobe. It is not possible to obtain a "pointy graph", which would allow good discrimination as much as is desired in both radius and distance. The best you can try to get is an ambiguity function with a narrow peak and with sufficiently low side lobes, see the figure below.
  
  
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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 is not without its 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 that would be required of 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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-In conclusion, the choice of a //pulse compression// system depends on the type of selected waveform and of the method of generation and processing. The primary factors influencing the selection of a particular waveform are usually the radar requirements of range coverage, doppler coverage, range and doppler sidelobe levels, waveform flexibility, interference rejection, and signal-to-noise ratio //(SNR)//. The //pulse compression// provides a tradeoff for realizing increased range resolution and, hence, greater clutter rejection. In applications where an insufficient doppler frequency shift occurs.+In conclusion, the choice of a //pulse compression// system depends on the type of selected waveform and on the method of generation and processing. The primary factors influencing the selection of a particular waveform are usually the radar requirements of range coverage, doppler coverage, range and doppler sidelobe levels, waveform flexibility, interference rejection, and signal-to-noise ratio //(SNR)//. The //pulse compression// provides a tradeoff for realizing increased range resolution and hence, greater clutter rejection. In application where an insufficient doppler frequency shift occurs, range resolution is the best way to see a target in clutter.
  
-=====test===== 
radar/pulsecompression.1528323075.txt.gz · Last modified: (external edit)