In a frequency-modulated continuous-wave (FMCW) radar employing a linear upsweep in frequency, the radar signal generator produces a phase modulated signal of the form
is the signal phase at time
is the radar carrier frequency,
is the sweep width, and
is the pulse repetition period. The instantaneous frequency of the transmitted signal is
In the course of the time interval
the radar frequency linearly varies between
In FMCW radar, the signal generated by the radar is split into two parts. A small portion is allowed to pass into the upper port of the radar mixer where it is used as a reference signal for detection of the echo signal. The other, larger portion of the generated signal passes out through the circulator and into the antenna. The circulator is a multiport, electronic device that only allows electrical signals to propagate in the clockwise direction. Use of a circulator allows the radar to share transmit and receive antennas. After exiting the antenna, this signal propagates into the air out to a target where it is reflected and returns to the antenna. It then goes back into the circulator, passes through in the preferred direction and into the horizontal port of the mixer. If the target is located at distance
from the radar, then the echo signal that comes into the mixer can be written in the form
is the propagation delay of the echo,
is the speed of light, and
is a factor that accounts for propagation losses, target reflectivity, and a variety of radar performance parameters.
The mixer multiplies the echo signal and the reference signal that came into the upper port of the mixer. This process produces sum and difference frequencies. The sum frequencies are on the order of twice the radar carrier frequency. The radar electronic circuits cannot respond to signals at this frequency. Thus only the difference frequencies pass out of the mixer. The signal that comes out of the mixer can be written
is the beat frequency of the target echo,
is the radar pulse modulation frequency, and
is the range resolution of the radar pulse.
In FMCW radar applications, targets are found by performing Fourier analysis on the mixer output signal. Using a Fourier transform with a rectangular window of integration extending over a time interval of length
leads to the following representation of the power in the mixer output signal:
is the analysis frequency and
is the power of the target echo. In terms of the standard radar equation, the target echo power
can be written
is the radar transmission power (W),
is the radar cross section
is the wavelength of the radar (m),
is the distance from the radar to the target (m), and
are the gains on a power scale of the transmit and receive antennas. For a single antenna system these gains are equal (
). Gain is usually expressed on a decibel scale. The target echo path is illustrated by the solid blue line
in the circuit diagram.
It is more effective to perform the Fourier analysis of the mixer output signal with a nonuniform window of integration in order to reduce side lobes in the spectral response. For the case of a Hamming window of integration
Use of the Hamming window of integration allows the detection of targets that are close together in range with large differences in echo level. Otherwise the 13 dB down side lobes (rectangular window) from the target with the larger echo will obscure the return from the target with the weaker echo.
The detection performance of the radar is limited by phase noise that propagates via the parasitic paths
illustrated as the dashed and solid red lines in the circuit diagram. The power spectrum of the phase noise of the signal generator is modeled via
It is measured in units of W/Hz. If the phase noise effects are included in the representation for the FMCW beat signal out of the mixer, then
is the phase of the phase noise at time
is the difference in travel time on the parasitic path (
) and the travel time on the reference path that goes directly from the signal generator to the mixer. If the parasitic path and the reference path have the same travel times then the phase noise cancels.
The frequency transfer function that models the process of obtaining a difference between two noise processes where one is a time-delayed version of the other is
The phase noise that limits radar performance is
is the loss in signal power measured in decibels on the parasitic path relative to the reference path.
Radar performance is ultimately limited by thermal noise. If
is the temperature of the radar electronics in degrees Kelvin (about 290 K at room temperature) and
is Boltzmann's constant
), then the power spectrum of thermal noise that limits radar performance is
where the noise factor
represents the decibel level increase of the radar thermal noise floor above the theoretical lower limit. An ideal (unobtainable) radar system would have a noise factor of 0 dB.
The signal-to-noise ratio for the radar echo is the ratio of signal power to the sum of all limiting noise effects. For a FMCW radar with two parasitic phase noise paths as shown, it can be written in the form
Reliable detection of a radar echo requires signal to noise ratios in excess of 10.
is usually expressed on a decibel scale.
 G. M. Brooker, "Understanding Millimeter Wave FMCW Radars," in First International Conference on Sensing Technology
, November 21–23, 2005, Palmerston North, New Zealand (G. Sen Gupta, S. C. Mukhopadhyay, and C. H. Messom, eds.), pp. 152–157.
 I. V. Komarov and S. M. Smolskiy, Fundamentals of Short-Range FM Radar
, Boston: Artech House, 2003.