In the previous chapters, the concept of the minimum detectable signal was introduced and was defined the concept of radar range in terms of statistics. In the sequel, some significant aspects of optimum filtering are illustrated.

The process of filtering consists of passing the signal through a linear system to modify it in a convenient manner. Here, we are interested in a particular kind of filtering in which the optimality criterion to be obtained is the maximisation of the signal-to-noise power ratio. In particular, in the case of the so-called matched filters, the output peak signal-to-noise power ratio, SNR, is maximised. 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. 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: $S{min}= SNR_{min}Fk_{b}T_{0}B_{n}$, 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.

• Suppose you have transmitted a rectangular pulse with amplitude equal to $T$. (See the figure below)

• 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 showed in figure (a), the receiver's output is expected to have a signal + noise like in figure (b).

The temporal shape of the useful signal in figure (b) is obtained at the end of the single-tuned filter, when in input you have a rectangular signal. Analyzing the phenomenon in the frequency domain, the useful signal (rectangle of unit amplitude and length $T$) has the following spectrum:

\begin{equation} X(f) = \frac{\sin (\pi T f)}{\pi f} \end{equation}

In the figure below is shown the superimposed white noise spectrum of the useful signal.

Now suppose we have a filter in reception $H(f)$, with a bandwidth equal to $\frac{1}{T}$. The response of the filter is shown in the figure below.

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 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

• At the instant $t = t_{0}$, the ratio signal-useful/noise is null;
• At the instant $t = t{1}$, the ratio signal-useful/noise is constant for $ \: \: t_{A} -T<t<t_{A}$

Now suppose to have a filter in the reception of type “single-tuned”, that has a band comparable with $\frac{1}{T}$. Consequently, the output signal will have the form in the figure below;

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. 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.

in the figure above, is shown a maximum, obtained through the ratio between a value of $S$ and the relative value of $N$.In conclusion, assigned a waveform with a spectrum $S(f)$, and a noise process, we must find a reception filter that is able to maximize the $SNR$ in the output. In case of the rectangular shaped pulse, it is possible to fix the type of filter and optimize the bandwidth. For example, can be considered an impulsive response;

\begin{equation} h(t) = \frac{1}{\theta}e^{\frac{-t}{\theta}}U(t)\end{equation}

• 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: $B=\frac{0.4}{T}$, 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. 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 instant time $t_{1}$ in which the signal $y(t)$ is the maximum noted value.

The maximum output power of the useful signal is:

\begin{equation}P_{s_{out}} = \left | y(t_{1}) \right |^{2} = \left | \int_{-inf}^{+inf}Y(f)e^{j2\pi ft_{1}}\,df \right |^2 \end{equation}

However in the Fourier Domain we have:

\begin{equation}Y(f)=S(f)H(f)\end{equation}

then the maximum power of useful ouput signal is:

\begin{equation}P_{s_{out}} = \left | \int_{-inf}^{+inf}S(f)H(f)e^{j2\pi ft_{1}}\,df \right |^2 \end{equation}

the power associated with the noise is equal:

\begin{equation}N_{out} = \int_{-inf}^{+inf} N(f) \left | H(f) \right |^{2}\,df \end{equation}

in which $N(f)$ is the spectral density of the noise.

Now we can calculate the ratio signal/noise: \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 $S/N$ 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}

If the process of noise is white with an even bilateral density equal to $N_{0}/2$, we have;

\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 }{\frac{N_{0}}{2} \int_{-inf}^{+inf} \left | H(f) \right |^{2}\,df } \end{equation}

To maximize the ratio we can use the inequality of Schwartz

\begin{equation} \left | \int_{D} A(f)B(f)\,df \right|^{2} \le \left ( \int_{D} \left| A(f) \right|^{2}\,df \right ) \left ( \int_{D} \left| B(f) \right|^{2}\,df \right ) \end{equation}

Consequently we have:

\begin{equation} \left( \frac{\tilde{S}}{N} \right)_{out} \le \frac{ \left ( \int_{-inf}^{+inf} \left| A(f) \right|^{2}\,df \right ) \left ( \int_{-inf}^{+inf} \left| B(f) \right|^{2}\,df \right ) } {\frac{N_{0}}{2} \int_{-inf}^{+inf} \left | H(f) \right |^{2}\,df } = \frac{2E}{N_{0}} \end{equation}

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 of a white process.

If we find the transfer function such as:

\begin{equation} \left ( \frac{\tilde{S}}{N}\right )_{out} = \frac{2E}{N_{0}} \end{equation}

namely

\begin{equation} \left ( \frac{S}{N}\right )_{out} = \frac{E}{N_{0}} \end{equation}

then the problem of the matched filter is solved.

Placing in the equation \begin{equation} \left ( \frac{\tilde{S}}{N}\right )_{out} \end{equation}

\begin{equation} A(f) = S(f)e^{j2\pi ft_{1}} \end{equation}

\begin{equation} B(f) = H(f) \end{equation}

ensure:

\begin{equation} H(f) = S^{*} \left( f \right ) e^{-j2 \pi f t_{1} G} \end{equation}

$G$ is a constant, that is assumed unitary, while $e^{-j2 \pi f t_{1} G}$ is a phase term (Delay factor that allows you to make the filter physically feasible)

In the domain of time the adjusted filter has the following expression:

\begin{equation}\begin{split} h(t) = \int_{-inf} ^{+inf} H(f) e^{j2 \pi ft} \,df = \int_{-inf}^{+inf} S^{*} \left( f \right ) e^{-j2\pi ft_{1}} e^{j2\pi ft} \,df = \\ = \left( \int_{-inf}^{+inf} \left[ S \left (f \right) e ^{j2\pi ft_{1} } \right ] e^{-j2\pi ft} \,df \right)^{*} = s^{*}\left(t_{1}-t\right) \end{split}\end{equation}

In the figure below (a), the signal is showed, and in the figure (b) the relative impulsive response of the matched filter is showed.

For the generic signal $ s(t) $, it is necessary to compute the convolution between $s(t)$ and $s^{*}(t_{1} -t)$ , that is the correlation between $s(t)$ and $s(t -t_{1})$.We need to correlate the received signal with its delayed reply of $t_{1}$ seconds, consequently the matched filter output is equal to:

\begin{equation} y(t) = \int_{-inf}^{+inf} s^{*}\left ( t-t_{1}+\theta \right ) s\left( \theta\right)\,d\theta \end{equation}

and coincides with $ R_{s}(t - t_{1})$ which is the $s(t)$ autocorrelation function of $t_{1}$ seconds. Consequently the matched filter, it is also called $\textbf{ correlation receiver}$.

In the case of non-white noise, the power of useful signal is equal to:

\begin{equation} P_{out} = \left| \int_{-inf}^{+inf} X(f)H(f) e^{j2\pi ft_{1}}\,df \right|^{2} \end{equation}

while the power of the noise is:

\begin{equation} N_{out} = \int_{-inf}^{+inf} \left| R(f) \right|^{2} \left | H(f) \right|^{2}\, df \end{equation}

placing

\begin{equation} P^{*}\left( f \right) = \int_{-inf}^{+inf} \frac{X(f)e^{j2\pi ft_{1}}}{R(f)} \end{equation}

\begin{equation} Q(f) = R(f)H(f) \end{equation}

it result:

\begin{equation} \left ( \frac{\tilde{S} }{N}\right )_{out} = \frac { \left | \int_{-inf}^{+inf}P^{*} \left ( f \right ) Q \left ( f \right ) \ ,df \right|^{2}} { \int_{-inf}^{+inf}\left| Q\left ( f \right ) \right|^{2}\,df} \end{equation}

using the inequality of Schwartz we have:

\begin{equation} \left ( \frac{\tilde{S} }{N}\right )_{out} \le \int_{-inf}^{+inf} \left | P \left ( f \right ) \right |^{2} \,df \end{equation}

remembering the definition of $P$ and doing a simple replacement, we have:

\begin{equation} \left ( \frac {\tilde{S} }{N}\right )_{out} \le \int_{-inf}^{+inf} \left | \frac{X \left ( f \right ) }{R \left ( f \right ) } \right|^{2}\,df \end{equation}

The sign of equality is valid only if: \begin{equation} Q \left ( f \right ) = GP \left ( f \right ) \end{equation}

where $G$ is constant, that is when:

\begin{equation} H \left ( f \right ) = H_{0} \left ( f \right ) = \frac{G}{\left | R \left( f \right ) \right|^{2}}X^{*} \left ( f \right ) e^{-j2\pi ft_{1}} \end{equation}

for $ H \left ( f \right ) = H_{0} \left ( f \right )$ , the ratio $ \left ( S/N \right )_{out}$ is maximum and it is equal:

\begin{equation} \left ( \frac{\tilde{S} }{N}\right )_{max} = \int_{-inf}^{+inf} \left | \frac{ X \left ( f \right ) }{ R \left ( f \right ) } \right|^{2} \,df \end{equation}

If \begin{equation} \left | R \left ( f \right ) \right |^{2} = cost = N_{0}/2 \end{equation} and placing:

\begin{equation} E = \int_{-inf}^{+inf} \left | X \left ( f \right ) \right |^{2} \, df \end{equation}

we obtain

\begin{equation} \left ( \frac {\tilde{S} }{N}\right )_{max} = \frac{2E}{N_{0}} \end{equation}

The optimum filter can be thought by a waterfall of two filters: One filter that makes uniform the spectral density of the noise, with a transfer function:

\begin{equation} H_{w} \left ( f \right) = \frac{1}{R \left ( f \right )} \end{equation}

is a matched filter of the signal, that is passed in the filter, whose transfer function is equal to:

\begin{equation} H_{s} \left ( f \right) = G \frac{X^{*} \left ( f \right) }{R^{*} \left ( f \right )} e^{j2 \pi ft_{1}} \end{equation}

This result leads to an important property of the matched filter for white noise, the maximum signal-to-noise power ratio is always twice the signal energy divided by the noise spectral density, and it is not dependent on the signal shape. The only problem is that we are not always able to make a matched filter for the given signal shape, because in some cases a non-causal filter could be realised. ^[2] 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.

Definition of the matched filter to the rectangular pulse, some examples of different types of filters

Suppose we have a carrier modulated by a rectangular pulse. To increase the energy associated with the pulse, it is possible to increase the amplitude of the signal and/or the duration $T$. Suppose we have an unitary amplitude, the waveform of rectangular signal is:

\begin{equation} s\left ( t \right ) = rect \left ( \frac{t-T/2}{T} \right) \end{equation}

The associated spectrum is:

\begin{equation} S \left ( \omega \right ) = \left ( \frac{sin( \omega T/2)}{ \omega/2} \right ) e^{-j\omega T/2 } \end{equation}

if we assume $t_{1} = T $, the frequency response of the matched filter to the rectangular pulse is:

\begin{equation} H \left ( \omega \right ) = \left ( \frac{sin( \omega T/2)}{ \omega/2} \right ) e^{-j\omega T/2 } \end{equation}

The output of the signal from the matched filter has the following spectrum:

\begin{equation} S_{out} \left ( \omega \right ) = \left ( \frac{sin( \omega T/2)}{ \omega/2} \right )^{2} e^{-j\omega T} \end{equation}

Practically the matched filter for the rectangular signal is impossible to realize because the filter band is unlimited. As a result, a low pass filter is used to detect the classic radar signal, whose response to a pulse is illustrated in the figure below.

This filter is an RC circuit, and therefore its band is equal to:

\begin{equation} B= \frac{1}{\pi RC} \end{equation}

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 ”.

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, with respect to the optimal ideal case (see the table below).

$T = $ duration of pulse

$B = $ band at $3 dB$

$B_{n} = $ noise band

For example, the loss of $SNR$ of the filter RC, so the optimum case, is equal to $0.89 dB $, while if use a filter of Bessel with three poly, there is a loss of only $ 0.5 dB $.

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.^[2] 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.

In the following paragraphs, the definitions of distance accuracy and radar resolution will be announced. Subsequently, there will be introduced basic concepts of sampling losses in surveillance radars, application of pulse compression, chirp signal with its complex representation and some solutions regarding the problem of lateral lobes.

Definitions

• The Range Accuracy indicates the degree of accuracy with which it is able to estimate the distance of the target

• Radar Resolution means the ability of a radar to distinguish between echoes that belong to two (or more) targets.

Suppose we send a rectangular pulse of duration $T$ at the instant $ t = 0 $, and after a time equal to $T_{R} $ we have the pulse echo. At the output of the reception filter, we have the situation shown in the figure below, in which $ V_{T} $ is the threshold voltage.

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) . 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.

\begin{equation} R = \frac{cT_{R}}{2} \end{equation}

Where c is the light speed

However, the signal which is detected is affected by noise, which limits not only its detection but also the estimation of the actual delay time as the latter is determined at the instant in which the received signal exceeds the threshold. With reference to the figure c, suppose that due to the noise the threshold is exceeded at point C, rather than at point B. Consequently, there is an error in measuring the distance of the target. The accuracy of the distance measurement is evaluated through the mean square error $\delta T_{R}$.

\begin{equation} \delta T_{R} = \sqrt { \overline{ \left ( \Delta T_{R} \right)^{2} } } \end{equation}

When $\Delta T_{R}$ is the measurement error; the bar indicates the statistical average. For high ratio $SNR$, indicating with $t_{r}$, the rise time, we can write:

\begin{equation} \Delta T_{R} \approx \frac{n(t)}{A/t_{r}} \end{equation}

it follows that

\begin{equation} \overline{ \left ( \Delta T_{R} \right)^{2}} = t^{2}_{r} \frac{\overline{n^{2}(t)}}{A^{2}} = \frac{N_{0}}{BA^{2}} = \frac{T}{2BE/N_{0}} \end{equation}

With the following expressions:

\begin{equation} t_{r} \approx \frac{1}{B} \end{equation}

\begin{equation} \overline{n^{2}\left ( t \right ) } \approx N_{0}B \end{equation}

\begin{equation} E = \frac{A^{2}T}{2} \end{equation}

B is the band of the filter IF, and E is the energy of the received signal, with duration T.

If we indicate with $SNR $ the ratio signal/noise in output of the matched filter equal to $ 2E/N_{0} $, the error RMS in distance $ \sigma_{R} $ is equal to:

\begin{equation} \sigma_ {R} = \frac{c}{2}\sqrt{\overline { \left( \Delta T_{R} \right )^{2} }} = \frac{c}{2} \sqrt{ \frac{T}{B \cdot SNR}} \end{equation}

we can see that $\sigma_{R}$, decreases with increasing of $B$ band and decreasing with the duration $T$. While for the rectangular pulse, the product BT , is of the order of unity, more complex waveforms can have much larger BT products than the unit. In this cases, fixing T , there are much wider bands of the rectangular pulse. If a measurement is made also on the falling front of the pulse, the distance is obtained by averaging the two measurements, ( the measure obtained from the rising front and the measure obtained from the falling front ), suppose that they are affected by two independent errors we have:

\begin{equation} \sigma_ {R} = \frac{c}{2} \sqrt{ \frac{T}{2B \cdot SNR}} \end{equation}

for $T \cong 1/B $ becames:

\begin{equation} \sigma_ {R} = \frac{c}{2B} \cdot \frac{T}{\sqrt{2 \cdot SNR}} \end{equation}

In the following, will be shown, what happens when the operation of measuring the distance and the delay time is carried out on the output signal of the matched filter. Consider the figure below, which shows the output of the matched filter to the rectangular pulse.

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 showed the procedure that gives the least average square deviation; it can be demonstrated that is equal to:

\begin{equation} \overline{ \left ( \Delta T_{R} \right )^{2} } = \frac{1}{ \beta \sqrt{\frac{2E}{N_{0}} } } \end{equation}

Where $\beta$ is the effective bandwidth defined as:

\begin{equation} \beta^{2} = \left ( 2 \pi \right )^{2} \frac { \int_{- \infty }^{+ \infty } f^{2} \left | X \left ( f \right ) \right |^{2} \,df } { \int_{- \infty }^{+ \infty} \left | X \left ( f \right ) \right |^{2} \,df} \end{equation}

$ 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 $SNR$. High accuracy is necessary for a tracking radar which only processes the pulses associated with the target being tracked.

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 .

It can be deduced that the precision in the measurement of the distance depends on the sampling step $ T_{c} $, usually we choose:

\begin{equation} T_{c} = T/n \end{equation}

where $ n \ge 1 $ is the number of samples for a pulse.

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 showed 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.

For reasons of complexity and values, it is not possible to improve the oversampling factor. The number of samples $ n $ is often reported according to the sampling losses in (dB), this is due to the fact that, statistically, the sample that exceeds the threshold does not fall in the area of maximum signal-to-noise ratio. The figure below, refers to the case in which we have a rectangular pulse with a Bessel filter, 3 poly; The curve A is applied to the fixed target, the curve B is for floating target, both for $ P_{D}= 0.80 $ and $P_{fa}=10^{-6}$.

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 showed in the figure below.

Due to the sampling and noise, the radar may not be able to resolve the two targets. In fact, considering the figure above, it can be verified that the samples A and B don't exceed the threshold, while the sample C, could overcome it. In these hypotheses, it is decided for the presence of a single target.

To solve the problem of too close targets, it is necessary to have a very “tight” pulse in the output of the matched filter. This means that the duration of the pulse $T$, must satisfy the requirements indicated below:

a) T must be small, so that discriminate between two or more pulses associated with distinct targets

b) T must be sufficiently large so that the energy of the received signal is sufficiently high.

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 .

The radar transmitters are normally in the saturation zone, the modulation is only ON-OFF wide. To address the waveform shape to improve resolution, it is better to modulate in phase or frequency . A typical signal is called “ chirp ”.

Applications of the chirp signal, complex envelope and problem analysis of lateral lobes.

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 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 ^[3].

On the basis of the concepts introduced in the previous paragraphs, now we will analyze the radar waveform that is of interest to this treatment: the chirp signal. A signal called 'linear chirp' is a real signal $ s(t) $, with narrow band with spectrum centered on frequency $f_{0}$, with constant amplitude and with quadratic function of time:

\begin{equation} a(t) = constant \: \: \: \: for \: \: 0<t<T \end{equation}

\begin{equation} \theta (t) = \frac{\mu t^{2}}{2} \end{equation}

where $\mu = \frac{2 \pi B}{T} $

$B$ = Is the band.

$T$ = Duration of the pulse.

\begin{equation} s \left ( t \right ) = 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 a Chirp signal with $ B= 1Mhz, T= 100 \mu s$ and $ f_{0} = 9Ghz $

The impulsive response of the matched filter of this signal is:

\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 .

The instantaneous frequency ( the first derivative before the argument, divided for $ 2 \pi $ ) of the chirp signal is:

\begin{equation} f_{i}(t) = f_{0} + \frac{1}{2 \pi} \frac{d\theta (f)}{dt} = f_{0} + \frac{\mu t}{ 2 \pi } \: \: \: \: for \: \: \: 0<t<T \end{equation}

whose trend is illustrated below, with $ B = \frac{\mu T}{2 \pi} $

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.

An expression of the output of the matched filter to the chirp signal, valid when $BT>>1$ , is obtained by assuming that the spectrum of the chirp signal has a constant amplitude $A_{0}$. Through some mathematical passages the output of the matched filter is:

\begin{equation} g(t) = \sqrt{ \frac{2 \mu }{\pi }} \frac{sin \left [ \frac{\mu t}{2} \left ( T - \left | t \right | \right ) \right ] }{ \mu t} cos(2 \pi f_{0} t) \: \: \: \: for \: \: \: -T<t<T \end{equation}

The width of the signal $g(t)$ to $-4 dB$ is about $T$ ; The product $BT$ is called compression ratio . The representation of the compressed pulse for $ B= 1Mhz $ and $T= 100 \mu$ , is shown in the figure below:

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 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 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.

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^[2]. In the following paragraphs, will be discussed methods to reduce the lateral lobes.

To simplify the analysis of a signal, we often recur to representation with the complex envelope . The complex envelope of signal is:

\begin{equation} u(t) = a(t)exp \left [ j \theta(t) \right ] \end{equation}

In the case of a chirp signal $ a(t) = constant \: \: \: \: $ for $ \: \: \: 0 < t < T $

\begin{equation} \theta (t) = \frac{\mu t^{2} }{2} \end{equation}

hence:

\begin{equation} u(t) = exp \left [ j \frac{\mu t^{2} }{2} \right ], \: \: \: \: for \: \: \: 0<t<T \end{equation}

The impulse response of the matched filter is:

\begin{equation} h(t) = k exp \left [ -j \frac{\mu t^{2} }{2} \right ], \: \: \: \: for \: \: \: \: 0<t<T \end{equation}

with $ k = \sqrt{\frac{2 \mu}{\pi}}$ (This value generates the frequency gain of the filter on the center frequency)

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^[2]. However, there are some techniques for reducing the level of the lateral lobes, namely:

• a) The technique allows to reduce the side lobes at the expense of an enlarged compressed pulse.
• b) The technique has the disadvantage of being more complex to realize and be sensitive to Doppler displacements.
• c) Frequency weighing is equivalent to inserting a filter that reduces the receiving side lobes.

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.

By indicating with $s(t)$ the transmitted signal (of spectrum $S(f)$ ) and with $ W(f)$ the frequency response of the filter for the reduction of the lateral lobes, the transfer function in reception is:

\begin{equation} W (f) \cdot S^{*}(f) \end{equation}

The output of the receiver (cascade filter with side lobe reduction filter), unless the constant factor, has a spectrum:

\begin{equation} G(f)= W(f) \: \: \: \: for \: \: \: \: \frac{-B}{2}<f<\frac{B}{2} \end{equation}

And null out of the chirp band.

From the theoretical point of view, the frequency weighing $ W(f) $ which allows to output the desired waveform $ g(t) $ is obtained from the inverse Fourier transform:

\begin{equation} W(f) = \int_{-inf}^{+inf} g(t)exp(-j2 \pi ft) \ ,dt \end{equation}

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.. It is possible to see in the image below about of Taylor and Dolph-Chebyshev's methods.

Instead, in the image below it is possible to see the enlargement of the compressed pulse weighed with respect to the compressed non-weighted pulse and the relative level of the lateral lobes with respect to the peak in cases where the weighing functions are: Dolph-Chebyshev for $\overline{n} = 5$ and Taylor for $\overline{n}=8 $

The figure below shows the Hamming function with, band $ B = 1Mhz $ .

Analysis of the chirp signal and some example of the blind zone.

We have observed that by using suitable methods of encoding the transmitted waveform it is possible to obtain advantages in terms of resolution in distance, using the signal output from the matched filter as the transmitted signal. In particular, a chirp signal is obtained, in comparison to the simple rectangular signal, we have an improvement of the resolution in distance with the same duration of the transmitted signal. In addition, there is less interceptibility of the transmitted pulse. The pulse “chirp” is long in time, the power that arrives, for example, on a target placed at $300km$, hardly detectable from the antennas of the war systems. The radar, on the other hand, manages to capture even modest powers because it has an antenna with a higher gain, and it uses the matched compression filter. The Low Probability of Intercept (LPI) project often leads to complicated encoding waveforms and Pulse Compression or Frequency-Modulated Continuous Wave (FMCW) solutions. When you have a long pulse you can not receive it for a long period of time and this leads to the existence of a blind zone. Consider the case in the figure below, in which, a pulse has been transmitted with a duration of $100 \mu s$.

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}

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).

To have the same resolution, the unencoded pulse duration must be in the order of $1/B$ where $B$ is the coded pulse band. For the first $T$ microseconds, the receiver is tuned to $f_{2}$, then to $ f_{1} $. Typical values are $ B = 2 MHz$, $\Delta T = 0.5 \mu s$ and $ T = 100 \mu s $. In the previous paragraph, it has been defined the coded signal of the type chirp. However, there are other methods of signal phase coding.

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 $\pi$ 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. ^[4]

A phase-coded signal (PSK) is relatively easy to generate. The following is an example of the case of binary encoding (2 phase values: $0$ and $\pi $ ). Suppose to transmit a T-length pulse composed of N sub-pulses (code elements) of duration $ \tau $. The local oscillator oscillates with a certain phase. At every instant $ \tau $, the oscillator output switches to a delay line or a phase-shifter corresponding to the desired phase value, for example, $ 180° $. In this way, the signal at the oscillator output is a phase modulated signal. The schema of a similar device is shown in the figure below.

A possible signal at the output of the schematized subsystem, in the figure above, is shown in the figure below. Using a number $n>1$ of phase shifters and a selector $n+1 \rightarrow 1$ (multiplex), the procedure is generalized to a code with $n+1$ phases.

The element that characterizes the phase modulation is the time interval, $\tau$, while the signal phase remains unaltered, said element o “chip” of the code. The analytical form of the signal $ s(t)$, modulated in phase in the discrete domain is:

\begin{equation} s(t) = \sum_{n=1}^N p_{n} \left ( t \right) e^{j \left ( \omega _{0}t + \theta _{n} \right)} \end{equation}

In which $p_{n} \left ( t \right )$ is equal to 1 if $t$ is included between $ (n-1) \tau $ and $n \tau$, $0$ elsewhere, $\theta _{n}$is the value of generic phase of of the chosen coding. A particular BPSK encoding is the one introduced by Barker where the encoding takes place over a time period of 2, 3, 4, 5, 7, 11, or 13 times. The longest is of length 13, this is a relatively low value for a practical pulse-compression waveform. When a larger pulse compression ratio is desired, some form of pseudorandom code is usually used ^[4].

The characteristic of these codes is that the autocorrelation function has a peak height of N (N = 2, 3, 4, 5, 7, 11 or 13) while the lateral lobes have an amplitude of 1. For example, consider the signal in the figure below, of duration $T = 5 \tau $.

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.

In the case of $N = 7$, the phase code is $ \left [ 0,0,0, \pi, \pi , 0, \pi \right ] $ (see the figure below of the autocorrelation function, Braker code, for N = 7 )

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} s[n] = ( e^{j0},e^{j0},e^{j0},e^{j \pi},e^{j0}, ) \end{equation}

The impulsive response of the numeric matched filter to the sequence is:

\begin{equation} h[n] = ( e^{-j0},e^{-j \pi},e^{-j0},e^{-j0},e^{-j0}, ) \end{equation}

As the phases assume only the values $ 0 \: \: and \: \: \pi $, one has that the coefficients of this matched filter are real and they are valid 1 or -1. The discrete signal at the output of the matched filter is shown in the figure below.

It is observed from the figure above, that there is a ratio between the peak and lobe amplitudes, which is equal to 5. The time-continuous signal at the output of the matched filter to the Barker code is the convolution of the signal in the figure above, with the waveform (time - continuous) representative of the sub-pulse. In general, Barker sequences have a peak/lobe ratio that is equal to N (number of signal elements). The Barker codes are only 7 and are shown in the table below.

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.

\begin{equation} s(t) = \sum_{n=1}^N p_{n}(t)e^{\omega _{0}t + \theta _{n}} \end{equation}

In which \begin{equation} \theta _{n} = \frac{1}{2}\mu n^{2} \tau ^{2} \end{equation}

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 used in the radar system:

• A) Frank's Codes
• B) Polyphase Codes
• C) Codes P1, P2, P3, P4
• D) Codes P (n, k)
• E) Complementary Codes

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} s_{t} (t) = ae^{\omega _{0}t + \phi (t)} \end{equation}

\begin{equation} s_{r} (t) = pae^{\omega _{0}t + \omega _{\Delta}t \phi (t) \psi} \end{equation}

where $\omega _{\Delta}$ it is the shift doppler, p is the attenuation term, $\psi$ is the displacement term.

In particular, the mismatch produces the following effects:

• A) The $SNR$ ratio at the output of the matched filter is less than the optimum value.
• B) There is a rise in the secondary lobe of the output signal from the matched filter.
• C) There is an enlargement of the output pulse from the matched filter and a consequent degradation of the radar resolution.

Suppose that the transmitted signal is unencoded. In the absence of noise and Doppler effect, it is expected to receive the signal in the figure below, in which the samples must have the same phase $\phi$.

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 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.

The effect of pulse compression on clutter depends on its nature. 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. 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)^[2].

Basic concepts regarding the ambiguity function and its properties.

Given a signal $s(t)$ (which can also be a sequence formed by the N pulses in the dwell time). The pulse response $ h(t) $ of the matched filter to that signal is, by definition, equal to

\begin{equation} h(t) = s* (t_{1} -t) \end{equation}

The output of the matched filter, if the received signal (which will be indicated with $ S_{D}(t )$ ) has been affected by the Doppler effect, in the absence of noise is:

\begin{equation} X \left ( t, f_{D} \right ) = \int_{-inf}^{+inf} s_{D}(\tau) \left ( t+ \tau \right ) \, d \tau = \int_{-inf}^{+inf}s (\tau )\left ( t + \tau \right ) e^{-j 2 \pi f_{D} \tau} \, d \tau \end{equation}

The last formula is called Ambiguity function According to the definition, we realize that it coincides with the autocorrelation function of the signal in the case of $ f_{D} = 0$

We can graph the function using the ambiguity diagram, the latter, represents the response of the matched filter to the signal for which is matched as well as to the doppler-frequency-shifted signals. Although it is seldom used as a basis for practical radar system design, in this case, we can graph the function $ X| t,f_{d}|^{2}$, and we obtain the figure below:

Some properties of the ambiguity function;

\begin{equation} a) \: \: \: \: |X(t,f_{D})| \le |X(0,0)| = E \end{equation}

\begin{equation} b) \: \: \: \: \int_{-inf}^{+inf}\int_{-inf}^{+inf}|X(t, f_{D})|^{2} \, dtdf_{D} =E^{2} \end{equation}

\begin{equation} c) \: \: \: \: |X(t,f_{D})| = |X(-t, -f_{D})| \end{equation}

Where $E$ is the Energy of the signal.

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.

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.

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 ^[4]. 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):

In the figure below, are illustrated the block diagrams of a radar Analog Pulse compression (Example: SAW with “chirp” )and Numeric Pulse compression (Example: Barker code) :

The Digital pulse compression techniques are used for both the generation and the matched filtering of radar waveforms. The digital generator uses a predefined phase-versus-time profile to control the signal. This predefined profile may be stored in memory or be digitally generated by using appropriate constants. The matched filter may be implemented by using a digital correlator for any waveform or else a “stretch” approach for a linear-FM waveform. Digital pulse compression has distinct features that determine its acceptability for a particular radar application. The major shortcoming of a digital approach is that its technology is restricted in bandwidths under $100 \: MHz$. Frequency multiplication combined with stretch processing would increase this bandwidth limitation. Digital matched filtering usually requires multiple overlapped processing units for extended range coverage. The advantages of the digital approach are that long-duration waveforms present no problem, the results are extremely stable under a wide variety of operating conditions, and the same implementation could be used to handle multiple-waveform types. ^[3]

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.