RadarLab

In radar technology the $Doppler Effect$ is using for the following tasks:

• Speed measuring;
• MTI - Moving Target Indication;
• in air-or space-based radar systems for precise determination of lateral distances.

The Doppler- Effect is the certain change in frequency or pitch when a sound source moves either toward or away from the listener, or when the listener moves either toward or away from the sound source. This principle, discovered by the Austrian physicist Christian , applies to all wave motion.

The apparent change in frequency between the source of a wave and the receiver of the wave is because of relative motion between the source and the receiver. To understand the Doppler effect, first, assume that the frequency of a sound from a source is held constant. The wavelength of the sound will also remain constant. If both the source and the receiver of the sound remain stationary, the receiver will hear the same frequency sound transmitted by the source. This is because the receiver is receiving the same number of waves per second that the source is producing.

if either the source or the receiver or both move toward the other, the receiver will perceive a higher frequency sound. For example, a sound source of frequency $f$ approaches you at a speed $ U_s$, its wavefronts will bunch together and you will hear a frequency $f ’$ which is higher than $f$. $( f ’ > f )$. This is because the receiver will receive a greater number of sound waves per second and denote the greater number of waves as a higher frequency sound. conversely, if the source and the receiver are moving independently the receiver will receive a smaller number of sound waves per second and will perceive a lower frequency sound i.e. $( f ’ < f )$.In both cases, the frequency of the sound produced by the source will have remained constant.

In other ways, the frequency of the Alarm on a fast-moving car sounds increasingly higher in pitch as the car is approaching than when the car is departing. Although the Alarm is generating sound waves of a constant frequency, and though they travel through the air at the same velocity in all directions, the distance between the approaching car and the listener is decreasing. As a result, each wave has less distance to travel to reach the observer than the wave preceding it. Thus, the waves arrive with decreasing intervals of time between them.

\begin{equation} F_d = \frac{2v}{\lambda } \end{equation}

• $ F_d $ = Doppler Frequency [Hz]
• $λ$ = wavelength [m]
• $v$ = speed of the wave-source [m/s]

This equation is valid, if the speed if the source of a wave is like the Radial speed ¹⁾.But the airplane usually flies in another direction than the direction towards to the radar.

• Green Vectors : Real speed
• Red Vectors : Radial speed
• Blue Vectors : Tangential speed

Only the radial speed is then also measured. However, this is different from the aim speed so that the following equation is valid:

\begin{equation} F_d=\frac{2v}{\lambda }\times cos\alpha \end{equation}

• $F_d$ = Doppler Frequency [Hz]
• $λ$ = wavelength [m]
• $v$ = speed of the aircraft [m/s]
• $α$ = angle between the direction of the transmitted/reflected signal and the direction of flight of the target

Derivation of the Doppler-frequency formula

It is well known in the fields of optics and acoustics that if either the source of oscillation or the observer of the oscillation is in motion, an apparent shift in frequency will result. This is the Doppler Effect and is the basic of CW radar. If R is the distance from the radar target, the total number of wavelength λ contained in a two-way path between the radar and the target is $\frac{2R}{\lambda }$. The distance $R$ and the wavelength λ are assumed to be measured in the same units. since one wavelength corresponds to an angular excursion of $2π$ radians, the total angular excursion of Φ made by the electromagnetic wave during its transit to and from the target is $\frac{4\pi {R}}{\lambda }$ radians. If the target is in motion, $R$ and the phase is continually changing. A change in $Φ$ with respect to time is equal to a frequency. This is the Doppler angular frequency $ ω_d $ given by

\begin{equation} \omega _d = \frac{d\varphi }{dt} =\frac{4\pi}{\lambda }\times \frac{dR}{dt} = \frac{4\pi {v_r}}{\lambda }=2\pi{F_d} \end{equation}

• $F_d$ = doppler frequency shift
• $v_r$ = relative(or radial)velocity of target wrt radar

The doppler frequency shift is

\begin{equation} F_d=\frac{2{v_r}}{\lambda }=\frac{2{v_r}F_0}{c } \end{equation}

• $F_0$ = transmitted frequency
• $c$ = $3\times 10^8$ [m/s]

This means in practice the Doppler- frequency occurs twice at a radar. Once on the way from the radar to the aim, and then for the reflected (and already afflicted by a Doppler-shift) energy on the way back.

What is Coherent Radar?

A type of radar that extracts additional information about a target through measurement of the phase of echoes from a sequence of pulses (or an extended observation interval, as in an FM–CW radar). The phase information may be used to improve the signal-to-noise ratio, to estimate the velocity of the target through the Doppler effect or to resolve the location of the target in a synthetic aperture radar. The word coherent means “in- phase” or maintaining a definite phase relationship with a certain reference waveform.

The transmitted pulse's of coherent radar have all defined phase angles to a reference. In other words, a pulsed radar system where the transmitted signal is phase-stable from pulse to pulse. The phase of a coherent signal at any point, relative to the reference signal is completely predictable.

As a transmitter different systems are used in radar.

In Coherent Radar Processing

One of the transmitting-system is the PAT (Power-Amplifier-Transmitter). In this case, the high-power amplifier is driven by a highly stable continuous RF source, called the Waveform generator ²⁾ Modulating the output stage in response to the PRF ³⁾ does not affect the phase of the driver/RF source. Assuming the RF is a multiple of the PRF (as is normally the case), each pulse starts with the same phase. Systems, which inherently maintain a high level of phase coherence from pulse to pulse, are termed fully coherent. It is taken to the necessary power with an amplifier such as (Amplitron, Klystron or Solid-State-Amplifier).

Note that phase coherence is maintained even if the PRF and RF are not locked together (provided the RF source is phase stable). As stated, it is common practice to lock the PRF to the RF phase and this assumption makes it easier to understand the concept of coherence.

• Note: Low Power oscillator and amplifier give same phase pulse to pulse and are a coherent system!

The most important benefit of this system is the ability to differentiate relatively small differences in velocity (which correspond to small differences in phase). This coherent target processing offers Doppler resolution/estimation and provides less interference and signal/noise benefits relative to non-coherent processing.

In Non-coherent Radar Processing

The phases of the transmitted signal are random from pulse to pulse. The phases of the echoes cannot be used to predict the range of the target. Another kind of the transmitting systems is the POT (Power Oscillator Transmitter) which is self-oscillating. When such a device is switched ON and OFF as a result of modulation by the rectangular modulating pulse, the starting phase of each pulse is not the same for the different successive pulses. The starting phase is a random function related to the startup process of the oscillator.

• Notice: Self-oscillating transmitter gives random phase pulse to pulse and is not coherent!

The envelope detector produces an output signal whose level corresponds to the envelope of the IF signal (linear detector, square law detector, or logarithmic detector) All frequency and phase information is lost.

What is the difference between coherent signals and non-coherent signals in a simple declaration?

Coherent systems need carrier phase information at the receiver and they use matched filters to detect and decide what data was sent, while non-coherent systems do not need carrier phase information and use methods like the square law to recover the data.

In other words, Coherency in signal processing is similar to correlation in statistics. In statistics, two random variables are correlated if there exists a linear relationship between the two. Perfectly coherent signals are signals such that one is the response of a linear system to the other applied signal. Hence there exists a linear system such that one signal is the input and the other signal is its output. Coherence is between $0$ and $1$. The higher the coherency is, the better one can explain the spectral content of the first signal by analyzing the spectral content of the other signal since a linear system is completely characterized by the system's associated Frequency Response Function (FRF). As a result, the formal definition is the cross spectrum of both signals divided by the square root of the auto spectra. Note that for the FRF we obtain that the FRF is given by the cross spectra of the input and output signal divided by the auto-spectrum of the input.

MOPA ( Master Oscillator Power Amplifier )

Generation of adequate RF power is an important part of any radar system. The radar equations showed that the transmitter power varies as the fourth root of the range if all other factors are constant. To double the range, The power has to be increased $16$-fold. buying range at the expense of power is costly; its therefore important that the best transmitter is selected for any particular application. Not only does a transmitter represent a large part of the initial cost of a radar system, but unlike many other parts of the radar, it requires a continual operating cost because of the need for prime power or fuel.

There are two basic transmitter configurations used in radar. One is the self-excited oscillator, exemplified by the magnetron. The other utilizes a low - power stable oscillator, which is in turn amplified to the required power level by one or more power amplifier tubes. An example is Klystron $Fig.7$ ⁴⁾ amplifier fed by a crystal-controlled, Frequency-multiplier chain, sometimes referred to as MOPA, an abbreviation for the master oscillator power amplifier. Both of those transmitter configurations encounter in the discussion of the MTI radar. The choice between the two is governed mainly by the radar application. Transmitters that utilize self-excited power oscillators are usually smaller than transmitters with master oscillator power amplifiers (MOPA).

The latter is more stable than self - excited oscillators and are usually capable of greater average power. Self-excited power oscillators, therefore, are likely to be found in applications where small size and portability are more important than stability and high power of MOPA.

In other words, the Master Oscillator is a very stable CW (Continuous Wave) crystal oscillator and constitutes the internal phase reference. It provides the coherent reference signal to the Phase Sensitive Detector and also through a frequency divider generates the system PRF in the Synchronizer. (more explanation in CW /Frequency-Modulated radars section)

Continuous Wave Radar

The type of radar which employs a continuous transmission, either modulated or unmodulated, has wide application. Historically, the early radar experimenters worked almost exclusively with continuous rather than pulsed transmissions ($sec$. Radar introduction). Two of the more important early applications of CW radar principle is the proximity VT fuze and the FM-CW altimeter. The CW proximity fuze was first employed in artillery projectile during World War and greatly enhanced the effectiveness of both field and antiaircraft artillery.

The CW radar is of interest not only because of its many applications, but its study also serves as a means for better understanding the nature and use of the Doppler information contained in the echo signal, whether in a CW or a pulse radar (MTI) application. In addition to allowing the received signal to be separated from the transmitted signal, the CW radar provides a measurement of relative velocity which may be used to distinguished moving targets from stationary objects or clutter.

consider the simple CW radar as illustrated by the block diagram of $ Fig.8$. The transmitter generates continuous (unmodulated ) oscillation of frequency $f_0$, which is radiated by the antenna. A portion of the radiated energy is intercepted by the target and is scattered, some of it in the direction of the radar, where it is collected by the receiving antenna.

If the target is in motion with a velocity $v_r$ relative to the radar, the received signal will be shifted in frequency from the transmitted frequency $f_0$ by an amount ± $f_d$ as given by the equation. The plus sign (+) associated with the Doppler frequency applies if the distance between target and radar is decreasing (closing target), that is, when the received signal frequency is greater than the transmitted signal frequency. The minus sign (-) applies if the distance is increasing (receiving target). The received echo signal at frequency $f_0 ± f_d$ enters the radar via the antenna and is heterodyned in the detector ( mixer ) with a portion of the transmitter signal $f_0$ to produce a Doppler beat note of frequency $f_d$. The sign of $f_d$ is lost in this process.

The purpose of the doppler amplifier is to eliminate echoes from stationary targets and to amplify the Doppler echo signal to a level where it can operate an indicating device. It might have a frequency - response characteristic similar to that of the figure.

The receiver of the simple CW radar of $Fig.8$ is in some respects analogous to a superheterodyne receiver. Receivers of this type are called Homodyne receivers, or superheterodyne receivers with Zero-IF. The function of the local oscillator is replaced by the leaking signal from the transmitter .such a receiver is simpler than one with a more conventional intermediate frequency since no IF amplifier or local oscillator is required.however the simpler receiver is not as sensitive due to the increased noise at the lower intermediate frequencies by Flicker effect ⁵⁾ .

Isolation between Transmitter and Receiver :

• Separate antennas are used for Transmitter and Receiver to reduce Transmitter leakage
• Local oscillator in the Receiver is derived from the Transmitter signal mixed with locally generated signal of frequency equal to that of the receiver IF.
• Transmitter leakage can occur due to Transmitter clutter also.

Reduction in flicker noise :

• Flicker effect noise reduces the receiver sensitivity of a CW Radar with zero-IF (Simple doppler radar). In order to increase the sensitivity and efficiency, we go for CW Radar with Non-zero IF.
• Doppler frequency usually falls in the audio or video frequency range which is more susceptible to flicker noise.
• Flicker noise is inversely proportional to frequency. So as we shift the Doppler frequency to IF flicker noise reduces.
• Super-heterodyne receiver with non zero IF increases the receiver sensitivity above $30$ dB

Receiver bandwidth :

• IF amplifier should be wide enough to pass the expected range of Doppler frequencies.
• Usually expected the range of Doppler frequencies will be much higher than the Doppler frequency. So a wideband amplifier is needed.
• But as a Receiver bandwidth in increased noise, increases and sensitivity degrade.
• And also the Transmitted signal bandwidth is not narrow.
• So Received signal bandwidth again increases.

When the Doppler-shifted echo signal is known to lie somewhere within a relatively wide band of frequencies, a bank of narrowband filters spaced throughout the frequency range permits a measurement of frequency and improves the SNR. The filters can be in either the RF, IF or video portion of the receiver.

The bandwidth of each individual filter is wide enough to accept the signal energy, but not so wide as to introduce more noise than need be. If filters are spaced with their half power points overlapped, the maximum reduction in the signal-to-noise ratio of a signal which lies midway between adjacent channels compared with the SNR at mid-band is $3$dB. The more filters used to cover the band, the less will be the maximum loss experienced, but the greater the probability of false alarm.

Direction of target motion :

In some application of CW radar, it is of interest to know whether the target is approaching or receding. This might be determined with separate filters located on either side of the intermediate frequency. If echo-signal frequency lies below the carrier, the target is receding; if the echo frequency is greater than the carrier, the target is approaching $ Fig.12$.

The sign of Doppler angular frequency shift $ω_d$ and the direction of the target's motion may be determined according to whether the output of channel $B$ leads or legs the output of channel $A$.

One method of determining the relative sign is:

If the output of channel B leads to the output of channel A, the Doppler shift is (Positive) Approaching Target. If the output of channel B leads to the output of channel A, the Doppler shift is (Negative) Receding Target.

\begin{equation} E_A(+) = k_2. E_0.cos(ω_d.t + Φ) , E_B(+) = k_2. E_0.cos(ω_d.t + Φ +π/2) \end{equation}

\begin{equation} E_A(-) = k_2. E_0.cos(ω_d.t - Φ) , E_B(-) = k_2. E_0.cos(ω_d.t - Φ -π/2) \end{equation}

Where:

• $E_0$ = amplitude of transmitter signal
• $ω_0$ = angular frequency of transmitter [rad/s]
• $Φ$ = a constant phase shift, which depends upon range of initial detection

Frequency Modulated CW radar

FMCW radar is capable to measure the relative velocity and the range of the target with the expense of bandwidth. An example of an amplitude modulation frequency is the pulse radar. By providing $timing$ $marks$ into the Transmitted signal the time of transmission and the time of return can be calculated. This will increase the bandwidth. More distinct the timing, more accurate the result will be and broader will the Transmitted spectrum. Here it is done by frequency modulating the carrier and the timing mark is the change in frequency.

Transmitted frequency increases linearly with time (solid line).Solid curve represents transmitted signal;Dashed curve represents echo.

The echo signal will return after a time $T = \frac{2R}{c}$ (dashed line). If the echo signal is heterodyned with a portion of the transmitter signal in a nonlinear element such as a diode, a beat note $f_b$ will be produced. If there is no Doppler frequency shift, the beat note is a measure of the target's range and $f_b = f_r$, If the rate of change of the carrier frequency is $f_0 $, the beat frequency is : \begin{equation} f_r = {f_0}T = \frac{2R}{c} f_0 \end{equation} In practical receivers, triangular frequency modulation is used.

If the frequency is modulated at a rate $f_m$ over a range $Δf$ the beat frequency is :

\begin{equation} f_r = \frac{2R}{c}2{f_m\Delta f} = \frac{4R{f_m}\Delta f}{c} \end{equation}

The reference signal from the transmitter is used to produce the beat frequency note. The beat frequency is amplified and limited to eliminate any amplitude fluctuations. The frequency of the amplitude-limited beat note is measured with a cycle counting frequency meter calibrated in distance, If the target is not stationary Doppler frequency shift will be superimposed on the FM range beat note and a wrong range measurement results

$f_b(up) = f_r - f_d$ $f_b(down) = f_r + f_d$

The beat frequency due to range frequency can be calculated as

\begin{equation} 1/2[f_b(up) + f_b(down)] = f_r \end{equation}

One-half the difference between the frequencies will yield the Doppler frequency. If there is “more than one target” the range to each target may be measured by measuring the individual frequency components by using a bank of narrowband filters. If the targets are moving the task of measuring the range of each becomes complicated.

FM CW Altimeter

To measure the height above the surface of the earth FM-CW radar is used as aircraft radio altimeter. Low Transmitted power and low antenna gain are needed because of short range. Since the relative motion between the aircraft and ground is small, the effect of the Doppler frequency shift may usually be neglected. And the frequency range is $4.2$ to $4.4$ Ghz (reserved for altimeters). Solid state Transmitted is used here. In general high sensitive super-heterodyne Receiver is preferred for better sensitivity and stability.

The output of the detector contains the beat frequency which contains doppler frequency and the range frequency. It is amplified to a level enough to actuate the frequency measuring circuits. The average frequency counter determines the range

$1/2[f_b(up) + f_b(down)] = f_r$

and The switched frequency counter determines the Doppler velocity. The averaging frequency counter is necessary for an altimeter since the rate of change of altitude is usually small.

In an altimeter, the echo signal from an extended target varies inversely as the square (rather than the $4$th power) of the range, because of greater the range greater the echo area illuminated by the beam. The low-frequency amplifier is a narrow band filter which is wide enough to pass the received signal energy, thus reducing the amount of noise with which the signal must compete. The average frequency counter is a cycle counter. It counts only absolute numbers. So there may be step errors or quantization errors.

Unwanted signals in FM altimeter:

1. The reflection of the transmitted signals at the antenna caused by impedance mismatch.
2. The standing-wave pattern on the cable feeding the reference signal to the receiver, due to poor mixer match.
3. The leakage signal entering the receiver via coupling between transmitter and receiver antennas. This can limit the ultimate receiver sensitivity, especially at high altitudes.
4. The interference due to power being reflected back to the transmitter, causing a change in the impedance seen by the transmitter. This is usually important only at low altitudes. It can be reduced by an attenuator introduced in the transmission line at low altitude or by a directional coupler or an isolator.
5. The double-bounce signal.

Moving Target Indicator

Using the principle of Doppler frequency shift in pulsed radar the relative velocity of the target can be determined. A pulse radar that utilizes the Doppler frequency shift as a means for discriminating moving from fixed targets is called a MTI (Moving Target Indication) or a pulse doppler radar.MTI is a necessity in high-quality air-surveillance radars that operate in the presence of clutter. Its design is more challenging than that of a simple pulse radar or a simple CW radar.

A CW radar is converted to pulsed radar by

The difference between simple pulse radar and pulse Doppler radar is that in pulse Doppler radar the reference signal at the receiver is derived from the transmitter, but in simple pulse radar, the reference signal at the receiver is from a local oscillator. Here the reference signal acts as the coherent reference needed to detect the Doppler frequency shift. The phase of the transmitted signal is preserved in the reference signal. Operation:

Let the CW oscillator signal be

\begin{equation} A_1.sin 2\pi{f_t} t \end{equation}

Then the reference signal is

\begin{equation} V_{ref}=A_2sin2\pi f_t t \end{equation}

Doppler shifted echo signal can be represented as,

\begin{equation} V_ {echo} = A_3sin[ 2\pi \left ( {f_t} {f_d} \right )t - \frac{4\pi {f_t}{R_0}}{c}] \end{equation}

The reference signal and the target signal are heterodyned in a mixer and the output is the difference frequency component

\begin{equation} V_ {diff} = A_4sin[2\pi {f_d}t - \frac{4\pi {f_t}{R_0}}{c}] \end{equation}

where

• $A_2$ =amplitude of received signal
• $A_3$ =amplitude of a signal received from a target at a range $R_0$

The difference frequency is the Doppler frequency. For stationary targets, $V_ {diff}$ is a constant. The voltages mentioned above are shown in the next $Fig$

Sample waveforms (bipolar)

Moving targets may be distinguished from stationary targets by observing the video output on an A-scope (amplitude vs. range). Echoes from fixed targets remain constant throughout, but echoes from moving targets vary in amplitude from sweep to sweep at a rate corresponding to the Doppler frequency. The superposition of the successive A-scope sweeps is shown in $Fig.23$ [$b$ to $e$] The moving targets produce, with time, a “butterfly” effect on the A-scope. It is not appropriate for display on the PPI.

Delay line cancelers: One method commonly employed to extract Doppler information in a form suitable for display on the PPI scope is with a delay-line canceler

The delay-line canceler acts as a filter to eliminate the DC component of fixed targets and to pass the AC components of moving targets.

Typical MTI radar (With Power Amplifier)

It differs in the way in which the reference signal is generated. The coherent reference is supplied by an oscillator called the Coho, which stands for a coherent oscillator. The coho is a stable oscillator whose frequency is the same as the intermediate frequency used in the receiver. The output of the coho FC is also mixed with the local-oscillator frequency $fl$.

The local oscillator must also be a stable oscillator and is called Stalo, for the stable local oscillator. The RF echo signal is heterodyned with the Stalo signal to produce the IF signal just as in the conventional super-heterodyne receiver. The characteristic feature of coherent MTI radar is that the transmitted signal must be coherent (in phase) with the reference signal in the receiver. This is accomplished by the coho signal. The function of the Stalo is to provide the necessary frequency translation from the IF to the transmitted RF frequency. Any Stalo phase shift is canceled on reception.

The reference signal from the coho and the IF echo signal is both fed into a mixer called the phase detector. Its output is proportional to the phase difference between the two input signals.

Triode, Tetrode, Klystron, Traveling-wave tube, and the Crossed-field amplifier can be used as the power amplifier. A transmitter which consists of a stable low- power oscillator followed by a power amplifier is sometimes called MOPA, which stands for master- oscillator power amplifier.

MTI radar (with power-oscillator Transmitter)

In an oscillator, the phase of the RF bears no relationship from pulse to pulse. For this reason, the reference signal cannot be generated by a continuously running oscillator. However, a coherent reference signal may be readily obtained with the power oscillator by remodifying the phase of the coho at the beginning of each sweep according to the phase of the transmitted pulse. The phase of the coho is locked to the phase of the transmitted pulse each time a pulse is generated.

A portion of the transmitted signal is mixed with the Stalo output to produce an IF beat signal whose phase is directly related to the phase of the transmitter. This IF pulse is applied to the coho and causes the phase of the coho CW oscillation to “lock” in step with the phase of the IF reference pulse.

The phase of the coho is then related to the phase of the transmitted pulse and may be used as the reference signal for echoes received from that particular transmitted pulse. Upon the next transmission, another IF locking pulse is generated to relock the phase of the CW coho until the next locking pulse comes along.

Delay Lines and cancelers

The simple delay-line canceler is limited in its ability to do all that might be desired of an MTI filter. The delay line must introduce a delay equal to the pulse-repetition interval. One of the advantages of a time-domain delay-line canceler, as compared to the more conventional frequency-domain filter, is that a single network operates at all ranges and does not require a separate filter for each range resolution cell. Frequency-domain doppler filter- banks are of interest in some forms of MTI and pulse-doppler radar.

Filter characteristics of the delay-line canceler

The delay-line canceler acts as a filter which rejects the DC component of clutter. Because of its periodic nature, the filter also rejects energy in the vicinity of the pulse repetition frequency and its harmonics. the video signal ($Eq.13$) received from a target at a range $R_0$ is

\begin{equation} V_1 = k.sin( 2πf_dt - Φ_0 ) \end{equation}

where $Φ_0$ = phase shift and $k$ = amplitude of video signal. The signal from the previous transmission, which is delayed by a time $T$ = pulse repetition interval, is

\begin{equation} V_2 = k.sin( 2π f_d(t+T) - Φ_0 ) \end{equation}

The output of a single delay canceller is

\begin{equation} V= \frac{V_1 }{2}-\frac{V_2 }{2} = k.sin \pi {f_d}T cos[2π f_d(t+\frac{T}{2}) - Φ_0 ] \end{equation}

It is assumed that the gain through the delay-line canceler is unity. The output from the canceler ($Eq.16$) consists of a cosine wave at the Doppler frequency $f_d$ with an amplitude ($k.sin \pi{f_d}T$): Thus the amplitude of the canceled video output is a function of the Doppler frequency shift and the pulse-repetition interval or PRF.the ordinate sometimes called the $visibility$ $factor$.

The response of the single-delay-line canceler will be zero whenever the argument $πf_d T$ in the amplitude factor of ($Eq.16$) is $0, π, 2π,.$, etc. or when

\begin{equation} f_d = \frac{n }{T}= n{f_r} \end{equation}

where $n$ = $0, 1, 2, ..$, and $f_r$ = pulse repetition frequency. The delay-line canceler, not only eliminates the DC component caused by clutter ($n = 0$), it also rejects any moving target whose doppler frequency happens to be the same as the prf or a multiple thereof. Those relative target velocities which result in zero MTI response are called blind speeds are given by

\begin{equation} v_n = \frac{nλ}{2T} = \frac{nλ{f_r}}{2} \end{equation}

where $n = 1,2,3,.$ and $v_n$ is the nth blind speed. If $λ$ is measured in [m], $f_r$ in [Hz], and the relative velocity in [knots], the blind speeds are

\begin{equation} v_n = \frac{nλ{f_r}}{102} \end{equation}

If the first blind speed is to be greater than the highest expected radial speed the $λf_r$ must be large

• Large $λ$ means larger antennas for a given beamwidth
• Large $f_r$ means that the unambiguous range will be quite small

So there has to be a compromise in the design of an MTI radar. The choice of operating with blind speeds or ambiguous ranges depends on the application

Two ways to mitigate the problem at the expense of increased complexity are:

• operating with multiple PRFs
• operating with multiple carrier frequencies

Double cancellation

The frequency response of a single-delay-line canceler $Fig.27$ does not always have as broad a clutter-rejection null as might be desired in the vicinity of D-C which limits their rejection of clutter and clutter does not have a zero width spectrum, Adding more cancellers sharpens the nulls.

The two-delay-line configuration of $Fig.28b$ has the same frequency-response characteristic as the double-delay-line canceler. The operation of the device is as follows. A signal $f(t)$ is inserted into the adder along with the signal from the preceding pulse period, with its amplitude weighted by the factor - 2, plus the signal from two pulse periods previous. The output of the adder is therefore

\begin{equation} f(t)- 2f(t+T)+ f(t+2T) \end{equation}

These have the same frequency response: which is the square of the single canceller response

\begin{equation} |v| = 4sin^2 (π f_d T) \end{equation}

Transversal filter

These are basically a tapped delay line, It is also sometimes known as a feed forward-filter, a non-recursive filter, a finite-memory filter. The weights $w_i$ for a three-pulse canceler utilizing two delay lines arranged as a transversal filter are $1, -2, 1$. The,frequency response function is proportional to $sin^2 π f_d T$, three delay lines whose weights are $1, -3, 3, -1$ gives a $sin^3 π f_d T$ response. This is a four-pulse canceler.

• Note the potentially confusing nomenclature. A cascade configuration of three delay lines, each connected as a single canceler, is called a triple canceler but when connected as a transversal filter it is called a four-pulse canceler.

To obtain a frequency response of $sin^n πf_d T$ ,the taps must be binomially weighted :

\begin{equation} w_i = (-1)^{i-1}\frac{n!}{(i-1)!(n-i+1)!} \end{equation}

Where $i = 1,2,3,.,n+1$

The transversal filter with alternating binomial weights is closely related to the filter which maximizes the average of the ratio

\begin{equation} I_C = \frac{(S/C)_{out}}{(S/C)_{in}} \end{equation}

where $(S/C)_{out}$ is the signal-to-clutter ratio at the output of the filter, and $(S/C)_{in}$ is the signal-to-clutter ratio at the input.

The ideal MTI filter should be shaped to reject the clutter at d-c and around the prf and its harmonics, but have a flat response over the region where no clutter is expected. That is, it would be desirable to have the freedom to shape the filter response, just as with any conventional filter. The ability to shape the frequency response depends to a large degree on the number of pulses used. The more pulses, the more flexibility in the filter design.

Unfortunately, the number of pulses is limited by the scan rate and the antenna beam width.

• Note: that not all pulses are useful, The first $n-1$ pulses in an n pulse canceller are not useful

The figure shows the amplitude response for ($1$) a classical three-pulse canceler with $ sin^2 π f_d T $ response, ($2$) a five-pulse “optimum ” canceler designed to maximize the improvement factor $3$ and ($3$) a 15-pulse canceler with a Chebyshev filter characteristic. The amplitude is normalized by dividing the output of each tap by the square root of

$\sum_{i=1}^{N} w_i^{2}$

where $w_i$ = weight at the $i$th tap

Shaping the frequency response

Non-recursive filters employ only feedforward loops. Feedforward (finite impulse response or FIR) filters have only poles (one per delay). More flexibility in filter design can be obtained if we use recursive or feedback filters ( also known as infinite impulse response or IIR filters )

These have a zero as well as a pole per delay and thus have twice as many variables to play with

IIR filters can be designed using standard continuous-time filter techniques and then transformed into the discrete form using Z transforms Thus almost any kind of frequency response can be obtained with these filters.

They work very well in the steady state case but unfortunately, their transient response is not very good. A large pulse input can cause the filter to “ring” and thus miss desired targets.

Since most radars use short pulses, the filters are almost always in a transient state

An alternative is to use multiple PRFs because the blind speeds (and hence the shape of the filter response) depends on the PRF and, combining two or more PRFs offers an opportunity to shape the overall response.

The closer the ratio $T_1$: $T_2$ approaches unity, the greater will be the value of the first blind speed. However, the first null in the vicinity of $f_d$ = $1 /T_1$ becomes deeper. Thus the choice of $T_1/T_2$ is a compromise between the value of the first blind speed and the depth of the nulls within the filter passband. The depth of the nulls can be reduced and the first blind speed increased by operating with more than two interpulse periods.

$Fig.36$ shows the response of a five-pulse stagger (four periods) that might be used with a long-range air traffic control radar.' In this example, the periods are in the ratio $ 25: 30: 27: 31$ and the first blind speed is $28.25$ times that of a constant PRF waveform with the same average period.

If the periods of the staggered waveforms have the relationship $n_1/T_1 = n_2/T_2 =.. = n_N/T_N$ where $n_l,n_2,...,n_N$ are integers, and if $v_B$ is equal to the first blind speed of a nonstaggered waveform with a constant period equal to the average period $T_{av} = (T_1 + T_2 +..+ T_N) / N $ , then the first blind speed $v_1$ is

\begin{equation} \frac {v_1}{v_B}= \frac{{n_1 + n_2 +..+ n_N}}{N} \end{equation}

Multiple PRFs can also be used with transversal filters, an example of $5$ pulse canceller with $4$ staggered PRFs

Digital Signal Processing

The convenience of digital means that multiple delay-line cancellers with tailored frequency-response characteristics can be readily achieved. And Most of the advantages of a digital MTI processor are due to its use of digital delay lines.

The down-converted signal is sampled by an A/D converter as follows:

• Note: The quadrature channel removes blind phases and the requirements for the A/D are not very difficult to meet with today’s technology.

Sampling Rate : Assuming a resolution ($R_{res}$) of $150$ m, the received signal has to be sampled at intervals of $c/2R_{res}$ = $1$μs or a sampling rate of $1$ Mhz

Memory Requirement : Assuming an antenna rotation period of $12$ s ($5$rpm) the storage required would be only $12$ Mbytes/scan.

Quantization Noise : The A/D introduces noise because it quantizes the signal.

The Improvement Factor can be limited by the quantization noise the limit being:

\begin{equation} I_{QN}= 20 log[ (2^N-1)\sqrt{0.75} ] \end{equation}

This is approximately $6$ dB per bit, A $10$ bit A/D thus gives a limit of $60$ dB In practice one or more extra bits to achieve the desired performance.

Dynamic Range: This is the maximum signal to noise ratio that can be handled by the A/D without saturation

Dynamic _ Range = $2^{2N-3} /k^2 $

• $N$ = number of bits
• $k$ = RMS noise level divided by the quantization interval (the larger k the lower the dynamic range but $k$<$1$ results in the reduction of sensitivity )

• Note: A $10$ bit A/D gives a dynamic range of $45.2$ dB.

Blind speed in an MTI radar

If the PRF is double the Doppler frequency then every other pair of samples can be the same amplitude thus it will be filtered out of the signal. By using both in-phase and quadrature signals, blind phases can be eliminated.

Digital filter banks and the FFT

A transversal filter with N outputs (N pulses and N - 1 delay lines) can be made to form a bank of N contiguous filters covering the frequency range from $0$ to $f_p$. Consider the transversal filter that was shown in $Fig.32$ to have N - 1 delay lines each with a delay time $T$ = $1/f_p $ . Let the weights applied to the outputs of the N taps be:

\begin{equation} W_{ik}= e^{-j}[2π(i-1)k /N] \end{equation}

And $i$ = $1,2,3..N$ and k is an index from $0$ to $N-1$.

The impulse response of this filter is:

\begin{equation} h_k(t)= \sum_{i=1}^{N}\delta \left [ t-(i-1)T) \right ]e^{-j2\pi (i-1)k/N} \end{equation}

The Fourier transform of the impulse response is the frequency response function

\begin{equation} H_k(f)=e^ {-j 2\pi f T }\sum_{i=1}^{N}e^{j2\pi(i-1)[fT-k/N]} \end{equation}

The magnitude of the frequency response function is the amplitude passband characteristic of the filter. Therefore

\begin{equation} |H_k(f)|=|e^ {-j 2\pi f T }\sum_{i=1}^{N}e^{ j2\pi(i-1)[fT-k/N]}| =| \frac{sin[\pi N(fT-k/N) ]}{sin[\pi (fT-k/N)]}| \end{equation}

For comparison, the improvement factor for an N-pulse canceller is shown in the next $Fig$.

• Note that the improvement factor of a two-pulse canceler is almost as good as that of the $8$-pulse doppler-filter bank. The three-pulse canceler is even better. ( Maximizing the average improvement factor might not be the only criterion used in judging the effectiveness of MTI doppler processors.)

Filter Banks can also be preceded by cancelers. The next figure shows the improvement factor for a three-pulse canceler and an eight-pulse filter bank in cascade, as a function of the clutter spectral width. ($a$) assumes uniform amplitude weighting ($- 13.2$ dB first sidelobe) and ($b$) shows the effect of Chebyshev weighting designed to produce equal sidelobes with a peak value of $-25$ dB.

Example Of An MTI Radar Processor

The Moving Target Detector (MTD) is an MTI radar processor originally developed by the MIT Lincoln Laboratory for the FAA's Airport Surveillance Radars $A S R$. The MTD processor employs several techniques for the increased detection of moving targets in clutter. Its implementation is based on the application of digital technology. It utilizes a three-pulse canceler followed by an $8$-pulse FFT doppler filter-bank with weighting in the frequency domain to reduce the filter sidelobes, alternate PRF's to eliminate blind speeds, adaptive thresholds, and a clutter map that is used in detecting crossing targets with zero radial velocity.

The input on the left is from the output of the $I$ and $Q$ A/D converters. The use of a three-pulse canceler ahead of the fi1ter: bank eliminates stationary clutter and thereby reduces the dynamic range required of the doppler filter-bank.

In the next $Fig$ the zero-velocity filter (No. 1) threshold is determined by the output of the clutter map. The thresholds for filters $3$ through $7$ are obtained from the mean level of the signals in the $16$ range cells centered around the range cell of interest. Both a mean-level threshold from the $16$ range cells and a clutter threshold from the clutter map are calculated for filters $2$ and $8$ adjacent to the zero velocity filter, and the larger of the two is used as the threshold. The advantage of using two PRF's to detect targets in rain is illustrated by the figure.

Limitation of MTI Performance

The improvement in the signal-to-clutter ratio of an MTI is affected by factors other than the design of the Doppler signal processor.

• MTI Improvement Factor ($I_C$) :

The signal-to-clutter ratio at the output of the MTI system divided by the signal-to-clutter ratio at the input averaged uniformly over all target radial velocities of interest. (discussed earlier)

• Subclutter Visibility ($SCV$):

The ratio by which a signal may be weaker than the coincident clutter and can be detected with the specified $P_d$ and $P_{fa}$. All radial velocities assumed equally likely.

$SCV = (C/S)_{in}$

• Clutter Visibility Factor ($V_{OC}$) :

The Signal to Clutter ratio after filtering that provides the specified $P_d$ and $P_{fa}$.

$V_{OC} = (S/C)_{out}$

Also, Clutter attenuation and Cancelation ratio which shall be explained more in the next parts.

Equipment instabilities

Pulse-to-pulse changes in the amplitude, frequency, or phase of the transmitter signal, changes in the Stalo or coho oscillators in the receiver, jitter in the timing of the pulse transmission, variations in the time delay through the delay lines, and changes in the pulse width can cause the apparent frequency spectrum from perfectly stationary clutter to broaden and thereby lower the improvement factor of an MTI radar.

Consider the effect of phase variations in an oscillator. If the echo from the stationary clutter on the first pulse is represented by $A cos ωt$ and from the second pulse is $A cos ( ωt + ΔΦ ) $. where $ΔΦ$ is the change in oscillator phase between the two, therefore the output of the two pulse filter is :

$ A cos\omega t - Acos(\omega t +\Delta \Phi ) = 2 A sin(\Delta \Phi /2) sin(\omega t +\Delta \Phi /2)$

the amplitude of the resultant difference is $2A sin \Delta \Phi /2$ ≈ $A \Delta \Phi$

Therefore the limitation on the improvement factor due to oscillator instability is

\begin{equation} I=\frac{1}{(\Delta \Phi )^2} \end{equation}

• Note : that if we need $I_C$ = $40$ dB ,The pulse to pulse phase variation has to be less than $0.01$ rad ($0.6º$).

Internal fluctuation of clutter

Although clutter targets such as buildings, water towers, produce echo signals that are constant in both phase and amplitude as a function of time, there are many types of clutter that cannot be considered as absolutely stationary. Echoes from trees, vegetation, sea, rain, and chaff fluctuate with time, and these fluctuations can limit the performance of MTI radar. Because of its varied nature, it is difficult to describe precisely the clutter echo signal.

Examples of the power spectra of typical clutter are shown in the next $Fig$ These data apply at a frequency of $1000$ MHz. The experimentally measured power spectra of clutter signals may be approximated by

\begin{equation} W(f) = |g(f)|^2 = \left | g_0 \right |^2exp\left [ -a(\frac{f}{f_0})^2 \right ] \end{equation}

where

• $ W( f )$ = clutter-power spectrum as a function of frequency
• $ g( f )$ = Fourier transform of input waveform (clutter echo)
• $f_0$ = radar carrier frequency
• $a$ = a parameter dependent upon clutter

(1) Heavily wooded hills, $20$ mi/h wind blowing ($a = 2.3 × 10^{17}$); (2) sparsely wooded hills, calm day ($a = 3.9 × 10^{19}$); (3) sea echo, windy day ($a = 1.41 × 10^{16}$)(4) rainclouds($a = 2.8 × 10^{15}$); (5) chaff ($a = 1× 10^{16}$).

The clutter spectrum can also be expressed in terms of an RMS clutter frequency spread $σ_c$ in [Hz] or by the RMS velocity spread $σ_v$ in [m/s],

\begin{equation} W(f)= W_0 exp\left (-\frac{f^2}{2\sigma _c^2} \right )= W_0 exp\left (-\frac{f^2\lambda ^2}{8\sigma _v^2} \right ) \end{equation}

where $W_0$ = $|g_0|^{2}$ , ${σ_c} = 2{σ_v}/λ$ , $λ$ = wavelength = $c/{f_0}$ , and $c$ = velocity of propagation. It can be seen that $a = c^2/8{σ_v}^2$ , The improvement factor can be

\begin{equation} I= \left ( \frac{S_o/C_o}{S_i/C_i} \right )_{ave}= \left ( \frac{S_o}{S_i}\right )_{ave} \times \frac{C_i}{C_o} = \left ( \frac{S_o}{S_i}\right )_{ave}\times CA \end{equation}

where $S_o/C_o$ = output signai-to-clutter ratio, $S_i/C_i$ = input signal-to-clutter ratio, and $CA$ = clutter attentuation. For a single-delay-line canceler, the clutter attenuation is

\begin{equation} CA= \frac{\int_{0}^{\infty } W(f) d(f)}{\int_{0}^{\infty}W(f)|H(f)|^2 df} \end{equation}

where $H ( f )$ is the frequency response function of the canceler. Since the frequency response function of a delay line of time delay $T$ is $exp (-j 2π f T)$ , $H(f)$ will be :

$H(f)$ = $ 1 - exp (-j 2π f T) $ = $ 2j sin ( π f T) exp ( -j π f T )$

And with assuming that $σ_c « 1/T$, the clutter attenuation is

\begin{equation} CA= \frac{\int_{0}^{\infty }W_0exp(-f^2/2\sigma _c^2)df}{\int_{0 }^{\infty} {W_0} exp (-f^2/2\sigma _c^2) 4 sin^2 \pi f T df}=\frac{0.5}{1-exp(-2\pi^2T^2\sigma _c^2)} \end{equation}

If the exponent in the denominator of the $Eq$ is small,

\begin{equation} CA= \frac{ {f_p}^2 }{ 4\pi ^2{\sigma _c}^2}= \frac{ {f_p}^2 \lambda ^2 }{ 16\pi ^2{\sigma _v}^2}= \frac{ {a}{f_p}^2 }{ 2\pi ^2{f_0}^2} \end{equation}

Where $f_p$ substituted for $1/T$ , The average gain $(S_o/S_{in})_{ave}$ of the single delay-line-canceler is $2$

\begin{equation} I_{1c} = \frac{ {f_p}^2 }{ 2\pi ^2{\sigma _c}^2}= \frac{ {f_p}^2 \lambda ^2 }{ 8\pi ^2{\sigma _v}^2}= \frac{ {a}{f_p}^2 }{ \pi ^2{f_0}^2} \end{equation}

And for the three pulse canceler, it is $6$

\begin{equation} I_{2c} = \frac{ {f_p}^4 }{ 8\pi ^4{\sigma _c}^4}= \frac{ {f_p}^4 \lambda ^4 }{ 128\pi ^4{\sigma _v}^4}= \frac{ {a^2}{f_p}^4 }{ 2\pi ^4{f_0}^4} \end{equation}

A plot of $Eq.32$ for the double canceler is shown in $Fig.39$ The parameter describing the curves is ${f_p}λ $. Example PRF's and frequencies are shown. Several “representative” examples of clutter are indicated, based on published data for $σ_v$, which for the most part dates back to World War II

Its a Plot of double-canceler clutter improvement factor [Eq.$32$] as a function of $σ_c$ = rms velocity spread of the clutter. The parameter is the product of the pulse repetition frequency ($f_p$) and the radar wavelength ($λ$).

In general

\begin{equation} I_{NC} = (\frac{ f_p }{ 2\pi \sigma _c}) ^ {2{N_l}} \times \frac{ 2^{N_l}}{{N_l}!} \end{equation}

Antenna scanning modulation

Since the antenna spends only a short time on the target, the spectrum of any target is spread even if the target is perfectly stationary:

The two-way voltage antenna pattern is

\begin{equation} G(θ) = G_0 exp [\frac{ - 2.776{θ}^2 }{ {θ_B}^2} ] \end{equation}

Dividing numerator and denominator of exponent by the scan rate

\begin{equation} S_a = G_0 exp [\frac{ - 2.776({θ/ \dot{\theta }_s})^2 }{ (\frac{θ_B}{\dot{\theta }_s})^2} ] \end{equation}

Letting $θ/ \dot \theta _s$ = $t$ ( the time variable ), and noting that ${θ_B}/ \dot{\theta _s}$ = $t_0$ ( the time on target ), the modulation of the received signal due to the antenna pattern is

\begin{equation} S_a = K exp [\frac{ - 2.776 {t^2} }{ {t_0}^2} ] = K_1 exp [ \frac{{-π^2}{f^2}{t_0}^2}{ 2.776 }] \end{equation}

where $K$ = constant. Since this is a Gaussian function, the exponent is of the form $ f^2 /2{σ_f}^2 $; where $σ_f$ = standard deviation. Therefore

\begin{equation} σ_f = \frac{1.178}{\pi {t_0}} \end{equation}

The power spectrum due to antenna scanning can be described by a standard deviation

\begin{equation} σ_s = \frac{1}{3.77 {t_0}} \end{equation}

To obtain the limitation to the improvement factor caused by antenna scanning. These are plotted in the next Figure.

\begin{equation} I_{1s} = \frac{{n_B}^2}{1.388} \end{equation}

\begin{equation} I_{2s} = \frac{{n_B}^4}{3.853} \end{equation}

When the MTI improvement factor is not great enough to reduce the clutter sufficiently..the clutter residue will appear on the display and prevent the detection of aircraft targets whose cross sections are larger than the clutter residue. Whereby setting the limit level $ L$, relative to the noise $ N$, equal to the MTI improvement factor $I$ or $L/N = 1$. If the limit level relative to noise is set higher than the improvement factor. clutter residue obscures part of the display and If it is set too low there may be a “ black hole ” effect on the display. The limiter provides a constant false alarm rate (CFAR) and is essential to usable MTI performance. Unfortunately, nonlinear devices such as limiters have side-effects that can degrade performance.

An example of the effect of limiting is shown in the Figure, which plots the improvement factor for two-pulse and three-pulse cancelers within various levels of limiting. The abscissa applies to a Gaussian clutter spectrum that is generated either by clutter motion with standard deviation $ σ_v$, at a wavelength $λ$ and a prf $f_p $, or by antenna scanning modulation with a Gaussian-shaped beam and $n_B$ pulses between the half-power beamwidth of the one-way antenna pattern. The parameter $C/L$ is the ratio of the RMS clutter power to the receiver-IF limit level.

The loss of improvement factor increases with increasing complexity of the canceler. Thus the added complexity of higher-order cancelers is seldom justified in such situations. The linear analysis of MTI signal processors is therefore not adequate when limiting & employed and can lead to disappointing differences between theory and measurement of actual systems.

More information is presented in Modern Radar System Analysis by David K.Barton chapter $6$.