JRG-77 Rev 1

The Linear FMCW Spectrum

-- Theory and Analysis

prepared by Mike Turley of DSTO

for discussion at the Fifth JRG 1A/1C/8B Meeting, London

April 2007

[Table of Contents]{.underline}

[1. Introduction 3](#introduction)

[2. LFMCW Theory 3](#lfmcw-theory)

[2.1. Single Chirp 3](#single-chirp)

[2.2. Multiple Chirps 5](#multiple-chirps)

[3. Spectral Analysis 7](#spectral-analysis)

[3.1. Fourier transform of a truncated repetitive signal
7](#fourier-transform-of-a-truncated-repetitive-signal)

[3.2. DFT Analysis 8](#dft-analysis)

[DFT of a single chirp 8](#dft-of-a-single-chirp)

[DFT of finite sequence of chirps 9](#dft-of-finite-sequence-of-chirps)

[DFT of infinite sequence of chirps
10](#dft-of-infinite-sequence-of-chirps)

[3.3. Spectrum Analyser 11](#spectrum-analyser)

[4. Conclusions 12](#conclusions)

Table of Figures:

[Figure 1 Example theoretical spectrum of a single chirp
4](#__RefHeading___Toc164008436)

[Figure 2 Example theoretical half spectrum of a single chirp
5](#__RefHeading___Toc164008437)

[Figure 3 Example frequency segment of multiple chirp modulation for T =
1/250 sec and N=16. 5](#__RefHeading___Toc164008438)

[Figure 4 Multiple chirp spectrum (N=8) 6](#__RefHeading___Toc164008439)

[Figure 5 Multiple chirp spectrum (N=128)
6](#__RefHeading___Toc164008440)

[Figure 6 Theoretical spectra for a single chirp and an infinite
sequence of chirps 7](#__RefHeading___Toc164008441)

[Figure 7 The Fourier transform of a truncated repetitive signal
8](#__RefHeading___Toc164008442)

[Figure 8 Example simulation of a single chirp.
9](#__RefHeading___Toc164008443)

[Figure 9 Conventional tapered DFT of finite sequence (8.4 chirps)
10](#__RefHeading___Toc164008444)

[Figure 10 DFT of single chirp as simulation of theoretical infinite
sequence 11](#__RefHeading___Toc164008445)

[Figure 11 Spectrum Analyser simulation using the STFT
12](#__RefHeading___Toc164008446)

[Figure 12 Spectrum Analyser Simulation
12](#__RefHeading___Toc164008447)

# Introduction

During the ITU-R Joint Rapporteurs Group JRG 1A-1C-8B fourth meeting at
Mainz, Germany, discussion took place on the spectrum of the Linear FMCW
waveform. In particular the following issues arose:

a)  should out-of-band emission formula be developed for a single chirp,
    multiple chirps or a continuum, and

b)  which case does the spectrum of a spectral analyser correspond to.

This paper provides a description of the LFMCW waveform and spectral
analysis methods.

# LFMCW Theory

The theoretical spectrum of the LFMCW waveform is discussed in a number
of JRG documents including JRG-70 and JRG-73. JRG-70 extends the
analysis to include flyback waveforms and 'non-continuous phase'
waveforms. Here we consider only the cases of a single chirp and that of
multiple chirps.

## Single Chirp

> Following JRG-73 the instantaneous frequency of a LFMCW waveform with
> bandwidth *B* and chirp duration *T* is:

  ------------------------------------------------------------- ---------
  ![](media/image1.wmf) ![](media/image2.wmf)                   \(1\)

  ------------------------------------------------------------- ---------

> where *t* is time and ![](media/image3.wmf) is the carrier frequency.
> The signal phase can be calculated as:

  ------------------------------------------------------------- ---------
  ![](media/image4.wmf) ![](media/image5.wmf)                   \(2\)

  ------------------------------------------------------------- ---------

> with the signal representation:

  ------------------------------------------------------------- ---------
  ![](media/image6.wmf)                                         \(3\)

  ------------------------------------------------------------- ---------

> The spectrum of this signal can be obtained from its continuous
> Fourier transform:

  ------------------------------------------------------------- ---------
  ![](media/image7.wmf)                                         \(4\)

  ------------------------------------------------------------- ---------

> letting ![](media/image8.wmf)and ![](media/image9.wmf)

  ------------------------------------------------------------- ---------
  ![](media/image10.wmf)                                        \(5\)

  ------------------------------------------------------------- ---------

> which can be solved in terms of Fresnel integrals C(*x*) and S(*x*):

  ------------------------------------------------------------- ---------
  ![](media/image11.wmf)                                        \(6\)

  ------------------------------------------------------------- ---------

> where ![](media/image12.wmf) and hence the power spectral density
> becomes:

  ------------------------------------------------------------- ---------
  ![](media/image13.wmf)                                        \(7\)

  ------------------------------------------------------------- ---------

> For example Figure 1 shows the spectrum calculated for B=10000 Hz and
> T=1/250 sec. The positive half spectrum is re-plotted with log
> frequency scale in Figure 2 together with overlaid 20 and 40 dB/decade
> roll-off curves. We note that the envelope spectrum of a single chirp
> has a 20 dB/decade sidelobe roll-off, and that for interest, its lower
> spectral envelope has a 40 db/decade roll-off.
>
> ![](media/image14.png){width="3.765277777777778in"
> height="2.189583333333333in"}

[]{#__RefHeading___Toc164008436 .anchor}Figure 1 Example theoretical
spectrum of a single chirp

> ![](media/image15.png){width="4.039583333333334in" height="2.2625in"}

[]{#__RefHeading___Toc164008437 .anchor}Figure 2 Example theoretical
half spectrum of a single chirp

## Multiple Chirps

> The spectrum of the repetitive form of the single chirp waveform may
> be derived from the sum of the spectra of time displaced single chirp
> waveforms. That is:

  ------------------------------------------------------------- ---------
  ![](media/image16.wmf)                                        \(8\)

  ------------------------------------------------------------- ---------

> where *n* denotes sweep index and *N* is the number of chirps, and the
> Fourier transform time-shift property has been used. The summation
> term of Equation 8 is shown in Figure 3 for the example of 16 chirps.
> The repetitive nature of the waveform is responsible for producing
> line spectra at intervals of 1/*T* Hz. These line spectra are of sinc
> form with width 1/*NT* Hz. As the number of chirps approaches
> ![](media/image17.wmf) the spectra becomes a set of delta functions
> with zero amplitude between the lines.
>
> ![](media/image18.png){width="5.793055555555555in"
> height="1.8451388888888889in"}

[]{#__RefHeading___Toc164008438 .anchor}Figure 3 Example frequency
segment of multiple chirp modulation for T = 1/250 sec and N=16.

> Multiplying this modulation by the single chirp spectrum (Figure 1)
> provides the multiple chirp spectrum as shown in Figure 4 & Figure 5.
> As the number of chirps increases the spectrum becomes discrete and
> the sidelobe roll-off asymptotes to 40 dB/decade. This is demonstrated
> in Figure 6 where the continuous spectrum from a single chirp is
> overlaid by the discrete spectrum of the infinite sequence. The
> infinite discrete spectrum can be obtained by sampling the single
> chirp spectrum at multiples of the chirp repetition frequency 1/*T*
> Hz. It may also be noted that the infinite chirp spectrum matches the
> lower envelope of the single chirp in the sidelobe zone.
>
> ![](media/image19.png){width="4.008333333333334in"
> height="2.2888888888888888in"}

[]{#__RefHeading___Toc164008439 .anchor}Figure 4 Multiple chirp spectrum
(N=8)

> ![](media/image20.png){width="4.020138888888889in"
> height="2.3194444444444446in"}

[]{#__RefHeading___Toc164008440 .anchor}Figure 5 Multiple chirp spectrum
(N=128)

> ![](media/image21.png){width="4.049305555555556in"
> height="2.259027777777778in"}

[]{#__RefHeading___Toc164008441 .anchor}Figure 6 Theoretical spectra for
a single chirp and an infinite sequence of chirps

# Spectral Analysis

We are also interested in the spectral analysis of numerically simulated
data or measurement data. Both simulation and measurement provide
avenues to analyse the spectra of a wider variety of waveforms.

## Fourier transform of a truncated repetitive signal

> To establish the Fourier transform of a finite repetitive sequence let
> us consider an infinite sequence of chirps, which has the
> corresponding discrete line spectra shown in Figure 7 (a & b). The
> temporal and spectral representations are known as a Fourier transform
> pair. Truncation of the sequence, Figure 7(c) equates to a convolution
> in the spectral domain between the discrete spectrum and the
> truncation function's point spread function. Figure 7(c & d) show the
> transform pair for a Hann window. The resulting truncated time
> sequence, Figure 7(e), has a continuous spectrum as in Figure 7(f),
> where for convenience the discrete spectrum is overlaid.
>
> ![](media/image22.png){width="1.9307086614173228in"
> height="0.36338582677165354in"}![](media/image23.png){width="1.897244094488189in"
> height="0.6507874015748032in"}![](media/image24.png){width="1.5133858267716536in"
> height="0.9720472440944882in"}![](media/image25.png){width="1.6574803149606299in"
> height="1.1909448818897639in"}![](media/image26.png){width="0.7846456692913386in"
> height="0.805511811023622in"}![](media/image27.png){width="0.8409448818897638in"
> height="0.4751968503937008in"}![](media/image28.png){width="0.908267716535433in"
> height="0.3716535433070866in"}![](media/image29.png){width="0.5818897637795276in"
> height="0.5708661417322834in"}![](media/image30.png){width="6.319444444444444e-2in"
> height="4.722222222222222e-2in"}

[]{#__RefHeading___Toc164008442 .anchor}Figure 7 The Fourier transform
of a truncated repetitive signal

## DFT Analysis

### DFT of a single chirp {#dft-of-a-single-chirp .unnumbered}

> The *M*-point discrete Fourier transform of the *M* point sequence
> *x*(*m*) is defined as:

  ------------------------------------------------------------- ---------
  ![](media/image31.wmf)                                        \(9\)

  ------------------------------------------------------------- ---------

> where the sample times are ![](media/image32.wmf), the sampling
> frequency is ![](media/image33.wmf), and the spectrum is calculated at
> the discrete frequencies![](media/image34.wmf). The DFT is typically
> calculated using a fast Fourier transform algorithm, where a
> convenient means to calculate intermediate frequency estimates is to
> append zero valued samples to sequence *x*. This increases the number
> of spectral indices *M* and is known as zero padding.
>
> An example single LFMCW chirp was numerically simulated and its
> spectrum calculated using an eight times zero padded DFT. The spectrum
> is shown in Figure 8 together with the envelopes of both the
> theoretical single and infinite chirp spectra overlaid. The simulation
> closely matches the theory apart from a deviation at low amplitude
> (see at high frequency), which is due to Nyquist aliasing. If required
> this may be reduced by sampling at a higher rate.
>
> ![](media/image35.png){width="4.048611111111111in"
> height="2.4256944444444444in"}

[]{#__RefHeading___Toc164008443 .anchor}Figure 8 Example simulation of a
single chirp.

### DFT of finite sequence of chirps {#dft-of-finite-sequence-of-chirps .unnumbered}

> Conventional DFT spectral analysis of a sequence, *x,* usually employs
> both the use of a taper function, *w,* to control sidelobe leakage and
> zero padding to interpolate spectral values:

  ------------------------------------------------------------- ---------
  ![](media/image36.wmf)                                        \(10\)

  ------------------------------------------------------------- ---------

> where *P* is the padding factor and *MP* is the number spectral
> points. This is demonstrated in Figure 9 for a chirp sequence of 8.4
> repetitions using *P*=4 and a Blackman-Harris 92dB taper function. The
> spectral envelope approximates that of the theoretical infinite chirp
> sequence, but does suffer from the Nyquist aliasing problem in this
> simulation. Under laid in green is the theoretical spectrum for the
> 8.4 chirp sequence assuming a uniform truncation function. This
> spectrum displays the poorer sidelobe roll-off rate at low levels, as
> discussed in Section 2.2 (see also Figure 4).
>
> ![](media/image37.png){width="3.661111111111111in"
> height="2.4368055555555554in"}

[]{#__RefHeading___Toc164008444 .anchor}Figure 9 Conventional tapered
DFT of finite sequence (8.4 chirps)

> We note that conventional DFT spectral analysis cannot be applied to
> the case of the LFMCW single chirp sequence as the taper modifies the
> waveform and the spectrum takes the form of an amplitude modulated
> LFMCW waveform.

### DFT of infinite sequence of chirps {#dft-of-infinite-sequence-of-chirps .unnumbered}

> The spectrum of an infinite repetitive sequence is a discrete spectrum
> with lines at multiples of the repetition frequency (1/T Hz). The
> discrete frequencies in the DFT equation (Equation 9) are calculated
> at:

  ------------------------------------------------------------- ---------
  ![](media/image38.wmf)                                        \(11\)

  ------------------------------------------------------------- ---------

> For the case where the signal chirp period is exactly a multiple of
> the sampling period ![](media/image39.wmf), we obtain:

  ------------------------------------------------------------- ---------
  ![](media/image40.wmf)                                        \(12\)

  ------------------------------------------------------------- ---------

> which produces frequency estimates at multiples of the repetition
> frequency. Hence the *M* point DFT of an *M* point single chirp
> sequence produces an estimate of an infinite chirp sequence. An
> example is shown in Figure 10 where the numerical simulation closely
> matches the theoretical curve for an infinite sequence, apart from the
> Nyquist aliasing.
>
> ![](media/image41.png){width="3.8055555555555554in"
> height="2.502083333333333in"}

[]{#__RefHeading___Toc164008445 .anchor}Figure 10 DFT of single chirp as
simulation of theoretical infinite sequence

## Spectrum Analyser

> The spectrum analyser is an instrument that scans a portion of the
> spectrum. At each scanned frequency the signal is passed through an IF
> stage which has a defined filter bandwidth known as the resolution
> bandwidth (RBW). When scanning an FM signal the filter response at any
> particular frequency becomes a function of time. Using the peak-hold
> function of the analyser will provide the spectral envelope of the
> signal.
>
> The process of the spectrum analyser may be considered by examination
> of the process in Figure 7, where the RBW filter response is
> equivalent to that of the point spread function. For example let us
> analyse an LFMCW signal with bandwidth 10000Hz and repetition
> frequency 50Hz. As a simple IF filter design consider a temporal
> domain Blackman-Harris taper truncation function (92dB sidelobes). For
> our purposes we require at least 60dB sidelobe filter response. For a
> RBW = 300Hz the taper truncation time is 1.90/300 =6.3ms (the 1.9
> value refers to the 3dB width for this taper \[Harris in Proc. IEEE
> V66 N1 1978\]). This example was simulated by the short time Fourier
> transform (STFT) analysis shown in Figure 11. The STFT has the
> advantage of estimating the spectrum at all frequencies for each time
> instant, unlike a true spectrum analyser instrument that is required
> to scan the frequencies over time.
>
> ![](media/image42.png){width="5.259722222222222in"
> height="2.6152777777777776in"}

[]{#__RefHeading___Toc164008446 .anchor}Figure 11 Spectrum Analyser
simulation using the STFT

Note the out-of-band emissions are mainly caused at the chirp
interfaces. The resulting spectrum analyser spectrum can thus be
obtained by applying a peak-hold function and is shown in Figure 12 for
various RBWs. The spectra are normalized to their maximum value, and are
found to approximate the theoretical spectrum. A true spectrum analyser
would not be subject to the Nyquist aliasing affects of this simulation,
provided an anti-aliasing filter was applied. We observe that the
in-band response is better for the lowest RBW, and that the recommended
![](media/image43.wmf)= 707Hz causes widening of the 40 dB bandwidth.

![](media/image44.png){width="4.045833333333333in"
height="2.6319444444444446in"}

[]{#__RefHeading___Toc164008447 .anchor}Figure 12 Spectrum Analyser
Simulation

# Conclusions

> In this report we have investigated the spectrum of the LFMCW
> waveform. Theoretical derivations were used to calculate the true
> spectrum for single, multiple and infinite chirp sequences. Then
> spectral analysis using DFT based techniques and a simulated spectrum
> analyser were conducted.
>
> The theoretical spectrum of a single LFMCW chirp may be calculated
> using Fresnel Integrals. The spectrum is continuous and has a sidelobe
> roll-off rate of 20 dB/decade. The spectrum of an infinite chirp
> sequence is a discrete modulation of the single chirp spectrum. The
> discrete lines are separated by the waveform repetition frequency (1/T
> Hz), and the out-of-band roll-off rate is 40 dB/decade. Note that
> between the lines the spectrum exists (continuous) but has zero valued
> amplitude.
>
> DFT spectral analysis of a single chirp also produces the 20 dB/decade
> roll-off provided the spectrum is also calculated at non-integer
> multiples of the repetition frequency. The DFT can also be used to
> calculate the spectrum of a finite chirp sequence. In this case if
> conventional DFT techniques of tapering and zero-padding are employed
> then the resulting spectral envelope matches that of the theoretical
> infinite spectrum. It is recommended that about eight or more chirps
> periods are sampled, and that a taper with at least 60 dB sidelobes is
> employed. As a special case, if the waveform repetition period is a
> multiple of the sampling period, then the DFT of a single chirp can be
> used to estimate the spectrum of the infinite sequence.
>
> An analysis and simulation of a spectrum analyser shows that the
> analyser provides an estimate of the infinite spectrum. However the
> scan rate and period, the choice of the IF filter bandwidth, and the
> filter sidelobe level are all significant.