The relationship between wavelength and frequency is as follows:

The speed of light is constant—Einstein proved this. Technically this is true only in a vacuum, but the effect of the medium such as our atmosphere can be ignored in radar discussions. What is happening in a radar system is that the frequency is modified by the process of being reflected by a moving object. Consider the transmission of a sinusoidal wave. The distance from the crest of each wave to the next is the wavelength, which is inversely proportional to the frequency. Each successive wave is reflected from the target object of interest. When this object is moving toward the radar system, the next wave crest reflected has a shorter round-trip distance to travel, from the radar to the target and back to the radar. This is because the target has moved closer in the interval of time between the previous and current wave crest. As long as this motion continues, the distance between the arriving wave crests is shorter than the distance between the transmitted wave crests. Since frequency is inversely proportional to wavelength, the frequency of the sinusoidal wave appears to have increased. If the target object is moving away from the radar system, then the opposite happens. Each successive wave crest has a longer round-trip distance to travel, so the time between arrival of receive wave crests is lengthened, resulting in a longer (larger) wavelength and a lower frequency. This effect becomes more pronounced when the frequency of the transmitted sinusoid is high (short wavelength). Then the effect of the receive wavelength being shorted or lengthened due to the Doppler effect is more noticeable. Therefore, Doppler frequency shifts are more easily detected when using higher frequency waves, as the percentage change in the frequency will be larger.

This effect only applies to the motion relative to the radar and the target object. If the object is moving at right angles to the radar, there will be no Doppler frequency shift. An example of this would be an airborne radar directed at the ground immediately below the aircraft. Assuming level terrain and the aircraft is at a constant altitude, the Doppler shift will be zero, even though the plane is moving relative to the ground. There is no change in the distance between the plane and ground.

If the radar is ground based, then all Doppler frequency shifts will be due to the target object motion. If the radar is vehicle or airborne based, then the Doppler frequency shifts will be due to the relative motion between the radar and target object. For example, if you are driving on the highway at 70 mph and an approaching police car is traveling at 50 mph, the radar will show a Doppler shift corresponding to 120 mph. The police radar will need to subtract the speed of the police car to display your speed.

This can be of great advantage in a radar system. By binning the receive echoes both over range and Doppler frequency offset, target speed as well as range can be determined. Also, this allows easy discrimination between moving objects, such as an aircraft, and the background clutter, which is generally stationary.

For example, imagine we have a radar operating in the X band at 10 GHz (λ = 0.03 m or 3 cm). The radar that is airborne, traveling at 500 mph, is tracking a target ahead moving at 800 mph in the same direction. In this case, the speed differential is −300 mph, or −134 m/s.

Another target is traveling head on toward the airborne radar at 400 mph. This gives a speed differential of 900 mph, or 402 m/s The Doppler frequency shift can be calculated as follows:

The receive signal will be offset from 10 GHz by the Doppler frequency. Notice that the Doppler shift is negative when the object is moving away (opening range) from the radar and is positive when the object is moving toward the radar (closing range).

The frequency response of an infinite train of pulses is similar, except that it is composed of discrete spectral lines in the envelope of the sinc function (Fig. 19.2). Also, the spectrum repeats at intervals of the pulse repetition frequency (PRF). We will forgo the mathematical derivation of this, but this is available in any engineering text on radar. This is not unlike a sampled signal, in which the frequency representation repeats at the sampling frequency interval (Fig. 19.3).

The important point is that this will impose restrictions on Doppler frequency shifts. To unambiguously identify the Doppler frequency shift, it must be less than the PRF frequency. Doppler frequency shifts greater than this will alias to a lower Doppler frequency and be ambiguous just as radar returns beyond the range of the PRF interval time are ambiguous, as they alias into lower range bins.

The Doppler frequency range placement will be some what arbitrarily determined by the digital downconversion of the received radar high-frequency carrier to baseband. Assuming downconversion of the carrier to 0 Hz, then the Doppler frequency effect will cause the target return signal to have a positive or negative offset, as computed below.

Doppler frequency detection is performed by using a bank of narrow digital filters, with overlapping frequency bandwidth (so there are no nulls, or frequencies that could go undetected). This is done separately for each range bin. Therefore, at each allowable range, Doppler filtering is applied. Just as the radar looks for peaks from the matched filter detector at every range bin, within every range it will test across the Doppler frequency band to determine the Doppler frequency offset in the receive pulse. This dramatically expands the amount of signal processing required. Rather than building many individual narrow-band frequency filters, the fast Fourier transform is used to perform spectral filtering across the spectral bandwidth of the PRF signal.

This gives a total Doppler range of 71.5 + 31.3 = 102.8 kHz. Unless the PRF exceeds 102.8 kHz, there will be aliasing of the detected Doppler rates and the associated ambiguities.

If we assume a PRF of 10 kHz from the previous chapter's example, we will clearly have Doppler ambiguities. Doppler ambiguities can be resolved using a number of methods.

In Fig. 19.4, the aliasing resulting in Doppler ambiguity is shown for a higher PRF of 80 kHz. If the PRF was 10 kHz, there would be many more Doppler ambiguities in the spectrum.

We have already discussed range and Doppler ambiguities. The PRF directly affects the size of the unambiguous zone. But ambiguities are not the only issue. Just as a range or Doppler measurement return can be outside the unambiguous zone and is aliased into the primary zone, so is all other returns and radar clutter. This can raise the noise floor of the radar to a degree that lower amplitude returns become obscured.

Sidelobe clutter is unwanted returns that are coming from a direction outside the mainlobe. Since the radar is not pointed in this direction, it is never a desired radar return signal. Sidelobe clutter is usually attenuated by 50 dB or more, due to the antenna directional selectivity or directional radiation pattern. A very common source of sidelobe clutter is ground return. When a radar is pointed toward the horizon, there is a very large area of ground area covered by the sidelobes in the negative elevation region. The large reflective area covered by the sidelobe can cause significant sidelobe returns despite the antenna attenuation.

Different types of terrain will have a different “reflectivity” that is a measure of how much radar energy is reflected back. It also depends on the angle of the radar energy relative to the ground surface. A related parameter, known the backscattering coefficient, has the angle incorporated into the coefficient and is therefore normalized over all angles. Some surfaces, such as smooth water, reflect most of the radar energy away from the radar transmitter, particularly at shallow angles. A desert would reflect more of the energy back to the radar, and wooded terrain would reflect even more. Man-made surfaces, such as in urban areas, would reflect the most of the energy back to the radar system.

This is one reason why Doppler processing is so important. Most targets are moving, and this is an effective method to distinguish them from the background clutter of the ground. Remember, the Doppler frequency of the ground will usually be nonzero if the radar is in motion. In fact, sidelobe Doppler clutter will vary by the elevation and azimuth angle, as the relative velocity will be equal to cosine θ, where θ is equal to the angle between direction on aircraft flight line and the ground. Different points on the ground will give different Doppler values, depending on how far ahead or off to the side of the radar-bearing aircraft that particular patch of ground is located. So Doppler sidelobe clutter will be present over a wide range of Doppler frequencies.

Mainlobe clutter is more likely to be concentrated at a specific frequency, since the mainlobe is far more concentrated (typically 3–6 degrees of beamwidth), so the patch of ground illuminated is likely to be far smaller and all the returns are at or near the same relative velocity.

A simple example can help illustrate how the radar can combine range and Doppler returns to obtain a more complete picture of the target environment (Fig. 19.5).

The diagram above illustrates unambiguous range and Doppler returns. This assumes the PRF is low enough to receive all the returns in a single PRF interval, and the PRF is high enough to include all Doppler return frequencies.

The ground return comes though the antenna sidelobe, known as sidelobe clutter. The reason ground return is often high is due to the amount of reflective area at close range, which results in a strong return despite the sidelobe attenuation of the antenna. The ground return will be at short range, essentially the altitude of the aircraft. In the mainlobe, the range return of the mountains and closing target are close together, due to similar ranges. It is easy to see how if just using the range return, it is easy for a target return to be lost in high-terrain returns, known as mainlobe clutter.

The Doppler return gives a different picture. First of all, the ground return is more spread out, around 0 Hz. The ground slightly ahead of the radar-bearing plane is at slightly positive relative velocity, and the ground behind the plane is at slightly negative relative velocity. As the horizontal distance from the radar-bearing plane increases, the ground return weakens due to increased range.

Notice that the Doppler return from the mountain terrain is now very distinct from the nearby closing aircraft target. The mountain terrain is moving at a relative velocity equal to the radar-bearing plane's velocity. The closing aircraft relative velocity is the sum of both aircraft velocity, which is much higher, producing a Doppler return at high velocity. The other target aircraft, which is slowly opening the range with radar-bearing aircraft, is represented as a negative Doppler frequency return.

Low PRF operation is generally used for maximum range detection. It usually requires a high-power transmit power, to receive returns of sufficient power for detection at a long range (remember, receive echo power levels are proportional to the range to the fourth power). To get the highest power, long transmit pulses are sent, and correspondingly long-matched filter processing (or pulse compression) is used. This mode is useful for precise range determination. Strong sidelobe returns can often be determined by their relatively close ranges (ground area near radar system) and filtered out. Disadvantages are that Doppler processing is relatively ineffective due to so many overlapping Doppler frequency ranges. This limits the ability to detect moving objects in the presence of heavy background clutter, such as moving objects on the ground. This can also be a problem for detecting low-flying aircraft because of ground terrain clutter at similar ranges in the mainlobe of the radar.

High PRF operation spreads out the frequency spectrum of the receive pulse, allowing a full Doppler spectrum without aliasing or ambiguous Doppler measurements. The clutter that is present in the spectrum is not folding or aliased from higher frequencies, which lowers the noise floor of the receive spectrum. A high PRF can be used to determine Doppler frequency and therefore relative velocity for all targets. It can also be used when a moving object of interest is obscured by a stationary mass, such as the ground or a mountain, in the radar return. The unambiguous Doppler measurements will make a moving target stand out from a stationary background. This is called mainlobe clutter rejection, or filtering. Another benefit is that since more pulses are transmitted in a given interval of time, higher average transmit power levels can be achieved. This can help improve the detection range of a radar system in high PRF mode.

Pulse delay–based ranging performance becomes very compromised in high-PRF operation. One solution is to use the high-PRF mode to identify moving targets, especially fast moving targets, and then switch to a low-PRF operation to determine range. Another alternative is to use a technique called FM ranging (Fig 19.6). In this mode, the transmit duty cycle becomes 100%—the radar transmits continuously. But it transmits a continuously increasing frequency signal and then, at the maximum frequency, abruptly begins to transmit at a continuously decreasing frequency until it reaches the minimum frequency resets to begin another cycle of increasing frequency. The frequency over time looks like a “sawtooth wave.” The receiver can operate during transmit operation, as the receiver is detecting time-delayed versions of the transmit signal, which are at a different frequency than current transmit operation. Therefore, the receiver is not desensitized by the transmitter's high power at the received signal frequency. Detection of what frequency is detected, and knowing the transmitter frequency ramp timing, can be used to detect round-trip delay time and therefore range. It is not as accurate as pulse-delay ranging using a matched filter but can provide ranging information nonetheless. Of course, the receive frequency will be affected by the Doppler frequency. On a rapidly closing target, the receive frequencies will be all offset by a positive f_Doppler, which can be measured by the receiver once the peak receive frequency is detected. The Doppler addition can be found as the receiver knows the peak frequency of the transmitter.

Medium-PRF operation is a compromise. Both range and Doppler measurements are ambiguous, but each will not be aliased or folded as severely as the more extreme low- or high-PRF modes. This can provide a good overall capability for detecting both range and moving targets. However, the folding of the ambiguous regions can also bring a lot of clutter into both range and Doppler measurements. Small shifts in PRFs can be used to resolve ambiguities, as has been discussed, but if there is too much clutter, the signals may be undetectable or obscured in both range and Doppler.

To track an identified target, repeated measurements are used over time. These measurements can be filtered to reduce measurement error, and the results of the filtering are fed back to control the measurement process. The radar system can respond by aiming of the mainlobe, by changing PRF, and by using measurements to anticipate future behavior of the target. For example, by estimating the target velocity and knowing the lag or latency in measurements, the radar can estimate the next position of target and have the mainlobe lead the target motion. This can also be done for the range binning and Doppler filtering. Also, if the radar itself is on a moving platform, such as an aircraft, the motion of the radar-bearing platform needs to be taken into account. This is referred to as platform stabilization.

Filtering of target measurements can be much more complex than the basic digital filtering discussed in previous chapters. The filtering may be recursive, where previous filter outputs are fed back, and will be adaptive, with gains and frequency cutoffs be varied in response to the measurement accuracy, degree of clutter, angle of antenna main beam, and other factors. There may be a number of independent or dependent filtering loops in operation. One loop may be tracking the range of target, by monitoring the range bins by detecting the comparative changes in adjacent range bin results, known as range gating. By doing this, the rate of change of the range can be coarsely estimated, which can lead to decision on when to switch to a high PRF to confirm with Doppler measurements. The antenna mainlobe may be electronically steered by making measurements at elevation and azimuths slightly above/below or side to side to estimate the azimuth and elevation of the highest return, to keep the mainlobe centered on a target. This is known as angle tracking, and this process also must account for motion of the radar platform. Note that this tracking activity may be a portion of the time, while another portion of time can be used for scanning or tracking of other targets.

In addition to tracking, there are often software-implemented algorithms to correlate the various measurements to particular targets. Target ranges, velocities, azimuths, elevations may cross over each other. These changes need to be interpolated into trajectory that can be matched to a specific target. Using radar digital signal processing followed by software-enabled target identification, tracking, monitoring, and classification, these functions can all be automated by the radar system. Higher level tracking by software can also allow for improvements in probability of detection and minimization of false alarms, as behavior of potential targets can be correlated and analyzed over longer time intervals than the radar measurement functions typically perform. This can allow the operator or pilot to more quickly understand the situation and spend more time on deciding how to respond.