Phase errors in the dechirped signal

The sources of signal phase (both ideal and errors) around the homodyne FMCW circuit are illustrated in Figure 3.4.

As discussed in Chapter 2, the phase of each of the dechirped echoes in the IF signal is the phase of the signal at the LO port of the mixer subtracted from that of the echo at the RF port.

Window function n a_n b_n c_n Uniform 0 1 0 0 Hann 0 0.5 0 0 1 −0.5 1 0 Hamming 0 0.54 0 0 1 −0.46 1 0 Cosine 0 1 0.5 −π/2 Blackman 0 (1−α)/2 0 0 1 1/2 1 0 2 α/2 2 0 Nuttall 0 0.355767 0 0 1 −0.487396 1 0 2 0.144232 2 0 3 −0.012604 3 0 Blackman-Harris 0 0.35875 0 0 1 −0.48829 1 0 2 0.14128 2 0 3 −0.01168 3 0 Blackman-Nuttall 0 0.3635819 0 0 1 −0.4819775 1 0 2 0.1365995 2 0 3 −0.0106411 3 0 Flat Top 0 1 0 0 1 −1.93 1 0 2 1.29 2 0 3 −0.388 3 0 4 0.032 4 0

Table 3.1: Coefficients for a selection of popular cosine family window functions and their PSFs _[72_]. The window is given by (3.5) and the corresponding spectrum (their PSF) is given by (3.8). Uniform, Hann, and Blackman-Harris window functions are shown in Figure 3.12.

X

LO RF

IF

Figure 3.4:The sources of the various additive parts of the signal phase in the homodyne FMCW radar.

The signal entering the LO port of the mixer has the following phase parts: φi(t) is the nominal phase of the signal.

2πR₀tε(t0)dt0 is the phase error in the signal leaving the source due to chirp nonlinearity. ε(t)is the frequency error in the chirp and t0 is a dummy vari- able.

η(t) is the phase error in the signal leaving the source not due to chirp non- linearity. This includes noise, group delay etc..

ηLO(t) is the phase error, consisting of noise and group delay etc., picked up

by the signal after being split into the reference (LO) arm and the transmit (tx) arm. This phase error only appears at the LO port of the mixer.

The total phase of the signal entering the mixer at the LO port is φ_LO(t), which is given by adding all of the separate parts that are listed and described immediately above:

φLO(t) =φi(t) +2π Z t

0

ε(t0)dt0+η(t) +ηLO(t). (3.9)

The same calculation can be done for the phase of the echo entering the RF port of the mixer. This differs slightly from the LO phase because there is

an additional delay,τ, caused by the the round-trip time of the echo.

The phase of the echo entering the RF port of the mixer has the following parts:

φi(t−τ) is the delayed copy of the nominal phase.

2πRt−τ

0 ε(t

0_)dt0 _{is the delayed copy of the phase error due to nonlinearity in}

the chirp.

η(t₋τ) is the delayed copy of the phase error in the signal due to group delay and noise etc. leaving the source not due to chirp nonlinearity.

ηtx(t−τ) is the delayed copy of the phase error in the signal due to group

delay and noise etc. between the coupler and being transmitted. This phase error only appears at the RF port of the mixer but is affected by the range of the reflector.

ηRF(t) is the phase error in the signal due to group delay and noise etc. in

the receiver. This phase error only appears in the RF port of the mixer and is not affected by the range of the reflector.

Combining the phase errors listed and described above gives the phase of the signal at the RF port,

φRF(t,τ) =φi(t−τ)+2π Z t−τ

0

ε(t0)dt0+η(t−τ)+η_tx(t−τ)+η_RF(t). (3.10) The phase of the IF signal leaving the mixer is given by

φIF(t) =φLO(t)−φRF(t,τ) +ηIF(t), (3.11)

where η_IF(t) is the phase error due to noise and group delay etc. introduced in the IF part of the signal processing.

Substituting forφLO(t)andφRF(t,τ)in (3.11) using (3.9) and (3.10) gives

the total phase of the dechirped signal, φd(t,R) =φi(t)−φi(t−τ)

| {z }

correlated nominal phase

+2π   Z t 0 ε(t0)dt0− Z t−τ 0 ε(t0)dt0   | {z }

correlated phase error due to chirp nonlin.

+ η(t)₋η(t₋τ)

| {z }

correlated phase error

+ηtx(t−τ) +ηRF(t) +ηIF(t)

| {z }

uncorrelated phase error

, (3.12)

The phase error in the dechirped signal from a reflector at range R is ξ(t,R), which is, in general, given by

ξ(t,R) = φ_d(t,R)

| {z }

total dechirped signal phase

−

φi(t)−φi(t−τ)

| {z }

correlated nominal phase

, (3.13)

and in this case is ξd(t,R) =2π   Z t 0 ε(t0)dt0₋ Z t−τ 0 ε(t0)dt0   | {z }

correlated phase error due to chirp nonlin.

+

η(t)−η(t−τ)

| {z }

correlated phase error

+ηtx(t−τ) +ηRF(t) +ηIF(t)

| {z }

uncorrelated phase error

. (3.14)

It can be seen from this example that some parts of the phase error vary with τ and that some do not. Of the parts that vary with τ, some correlate whilst others do not.

For the range-dependent uncorrelated phase errors, the PSF will vary with range because different regions of the amplitude and phase errors span the spectral analysis region. However, this is unlikely to make a significant differ- ence to the PSF as a function of range.

The variation inτ has particular implications for the correlated phase er- rors because the echoes and transmitting copies interfere in a way that signif- icantly modifies the resultant phase error in the signal.

In the case of chirp nonlinearity, the mixing of the delayed and undelayed copies of the signal results in the net phase error scaling proportionally to the range of the reflector. In the case of noise-like phase errors, the result is a modified phase error spectrum, which will be discussed in more detail in Chapter 6.