- Radar phase measurement accuracy

Phase is one of the basic measurements that can be made on radar signals. Because measured phase is used in deriving several important radar quantities, it is necessary to determine the accuracy of phase measurements. How can we measure phase error?

With the radar signal broken into real and imaginary components, we can note that the tangent of the phase of the signal is just the ratio of the imaginary over the real component, that is

tan(f) = Imaginary / Real

In radar terminology, which relates to the phase of a signal relative to some reference frequency, the real part is called inphase, and the imaginary part is called quadrature. The quadrature refers to the 90 degree phase shift, one quarter of a circle. In radar terminology, then,

tan(f) = Quadrature / Inphase

Now that we have a definition for phase, we can look at the signal and the noise.

We will establish a signal vector with power S. The amplitude of the vector is square root of S. We will set the signal so that it lies coincident with the inphase axis. Although this does not appear to be general, this is just a rotation of coordinate systems and will yield identical results independent of signal vector angle. It is much easier to visualize if the signal vector is fixed, and the coincidence of the signal vector with inphase enables easy understanding.

Noise is only a little more difficult to visualize. The key to noise is to know that narrowband noise can be represented by putting half the noise power N inphase and half in quadrature, so that each has power N/2. The inphase and quadrature noise vectors are independent of each other, and are identically distributed Gaussian random variables. The standard deviation of the noise in each channel is square root(N/2).

The inphase component of the noise does have an effect on the phase, but it is a second order effect. For any SNR that would be used in radar detection, the inphase noise has no impact on the phase measurement. Since we are concerned with phase only in this problem, we will drop further consideration of the inphase noise for this problem only. With this, the phase can now be approximated as

tan(f)» Quadrature Noise/ Signal

s_noise² = N

s_I² = s_Q² = N/2

. Signal =

For f small, tan(f)~ f. This is equivalent to S>>N. Because this is required for detection, the approximation is valid in all cases of interest.
To use the equation properly, note the following.

• SNR is signal power / noise power, or S/N.
• SNR in this equation is not in decibel units, it is simply a ratio.
• The phase error is in radians. It can be converted to degrees in the standard manner.
• The equation tells us the limit on achievable phase measurement accuracy. It does not guarantee that we will achieve this.

Problems

1. Draw a graph of the phase accuracy limit (in degrees) as a function of SNR (dB).
2. What SNR is required to get a 1° phase error?
3. If there are five independent phase measurements averaged, what SNR is required for each measurement to get a 1° phase error?
4. What is the formula for the best possible standard deviation of the difference between two phases (assume identical measurement conditions)?