# A New Demodulation and Modulation Method  Designed for FMCW Radar

Authors: Wei Shen; Biyang Wen
Journal: Journal of Electrical and Computer Engineering
 (2010)

---

## Abstract

An efficient demodulation method
designed for FMCW (Frequency-Modulated Continuous Wave)
radar is presented. It is a kind of modified DFT (IDFT) algorithm;
the spectrum segment of interest can be easily extracted from
the original signal without calculating the whole DFT/FFT. It
provides fast demodulation and extraction of desired frequency bands in our HFSWR (High-Frequency Surface Wave Radar)
system. The proposed approach enhances the performances of
radar system and reduces the computing complexity. The new
structure could also be inversely used for signal modulation. And
also arbitrary sampling rate conversion could be achieved with
the combination of forward and backward structure.


---

## Body

## 1. Introduction

FMCW is a radar system where a known stable frequency continuous wave radio energy is modulated by a triangular modulation signal so that it varies gradually and then mixes with the signal reflected from a target object with this transmit signal to produce a beat signal. With the advent of modern electronics, the use of Digital Signal Processing is used for most detection processing. The beat signals are passed through an Analog-to-Digital converter, and digital processing is performed on the result [1]. Fast calculating of demodulation and target detecting/tracking is an important issue in radar system design. In order to get higher Signal-to-Noise Ratio, high-speed sampling is used during Analog-Digital Converting. However, high sampling rate brings large bound of data for processing, which reduces performance of the real-time processing. SRC (Sampling Rate Conversion) method is widely used to isolate the desired channel (in frequency domain) in the communication systems [2]. Many different Digital Down-Converter (DDC) structures have been implemented, and the programmable part of decimation structure includes FIR filter, comb filter, and so on. The SRC turns out to be mainly a problem of designing appropriate filters and structures implement [3]. In this paper, combining with the principle of FMCW radar demodulation and the receiver structure, an efficient demodulation structure is proposed, and further extensions are also presented.


## 2. Signal Processing in FMCW Radar System

Frequency-Modulated Continuous Wave mechanism has been widely used in modern radar system. The waveform parameter of the mechanism is described as follows:(1)xT(t)=ej(2πfct+παt2),-Ts2<t<Ts2,
where fc is radar carrier frequency, α is the sweep rate, and Ts is the sweep repetition period; the received signal is a delayed (time td) and scaled (factor A ) replica of the transmit signal:(2)xR(t)=Aej(2πfc(t-td))+πα(t-td)2).The time delaytd can be extracted from xT(t) and xR(t), by the demodulation x(t)=xR(t)⊗xT(t), which is a simple function of the target range and velocity [4, 5]:(3)td(p,t,v)=F{x(t)}=2R(p,t,v)c=2c[Ro+v(pTr+t)],
where Ro is the initial target range, v is the velocity, c is the speed of the light, and p is the sweep number. Figure 1 gives the sketch of transmit and receive signal of FMCW system.Figure 1
Sketch of transmit and receive signal of FMCW.As shown by Barrick, and others [4, 6, 7], the frequency of the demodulated signal x(t) is approximately proportional to target's delay time τo by(4)Δf=ατo.
The range of specific targets can be extracted from the frequency of demodulated signal x(t). ADC is used to generate the discrete sequence x[n]; and according to sampling theory, in normal situation, if sampling rate fs is much higher than the band of desired signal (0~Δfmax), the higher Signal-to-Noise Ratio can be achieved [2]. In HF radar measurement, the corresponding maximum frequency offset Δfmax (calculated from the detecting physical limitation) is only several hundred Hz [4]. But the radar system operates in high-frequency band. The noise outside the receiver is dominant other than noise inside the receiver. So an appropriate sampling clock is determined according to outside noise level and the SNR we can get.


## 3. Principle of the Fast Demodulation in FMCW

As discussed above, the frequency information can be extracted from Fourier Transform (FT) ofx[n].  x[k] may be given by(5)x[k]=DFT{x[n]}=∑k=0N-1x[n]e-j2π(kn/N).
We define WN=e-j2π/N;  N points are sampled in each repetition period. An evaluation of (5) using Fourier transform (FT) would result in N points in frequency domain. But according to the maximum Δf offset related to the maximum detecting range (as we have discussed above). Frequency resolution of FT result is δF=1/Ts [8], from the maximum frequency offset and frequency resolution. The desired frequency points are determined. The first M points are enough for target detecting and further signal processing. Normally, M is set to 2Z (Z is positive integer; appropriate M is selected also according to M-point FFT block in hardware), so we have M points, and Q=N/M,M≪N. If the processor is very powerful and with enough storage memory, surely, it can process the whole sequence with once FFT calculation. But usually, the system needs to process multichannel received signals and some further processing including target detecting and tracking. So the efficiency of the processing must be considered.The new structure is described as follows: we expand   (5)(6)X(k)=x(0)WNk·0+,…,+x(Q-1)WNk·Q-1+x(Q)WNk·Q+,…,+x(2·Q-1)WNk·(2·Q-1)+x[(M-1)Q]WNk·(M-1)Q+,…,+x(MQ-1)WNk(MQ-1).
We get the sum of each column, for example, the sum of qth column (q∈[0,1,…,Q-1]):(7)Xq=x(q)WNkq+,…,+x[(M-1)·Q+q]WN[(M-1)·Q+q]=∑m=0M-1x(m·Q+q)WN(k(m·Q+q))=∑m=0M-1x(m·Q+q)WNk(m·Q)WNkq.
According to the Fourier transform formula,(8)Xq′=DFT{x[m·Q+q]}=∑m=0M-1x(m·Q+q)WMkm,WMkm=e-j2πkm/M=e-j2πkm/(N/Q)=WNk·(m·Q).
From formulas (7), (8), and (9), we will get(9)Xq(k)=Xq′(k)WNk×q,X0′′=∑q=0Q-1Xq(k)=∑q=0Q-1Xq'(k)WNk×q=X(k){k=1,2,…,M}.If we separate the result of total FT ofx[n] into Q segments, we get(10)X0′′=X(k){k=1,2,…,M},⋮Xp′′=X(k){k=p×M+1,p×M+2,…,(p+1)×M},⋮XQ-1′′=X(k){k=(Q-1)×M+1,(Q-1)×M+2,…,Q×M},
where p is a value of (0,1,2,…,Q-1), the Factor is defined as WNk×q(q=0,1,2,…,Q-1;k=1,2,…,M), and the output is X0′′. If we define(11)WNk×q(q=0,1,…,Q-1;k=p×M+1,p×M+2,…,(p+1)×M).
Then we can get Xp′′, which is the pth segment of the total FT (as shown in (10)). Figure 2 shows the block diagram of the principle of the proposed structure. X0′′ is the result of demodulation. So the N-point Fourier transform of sequence x(n) can be replaced by the sum of a relative shorter M-point FFT which will be multiplied by a series of factor WNk×q. So the analytic frequency band is converted to Fs/2Q, which meets the need of the echo signal processing as analyzed in Section 2. If p is defined as some certain value (p≠0), the corresponding frequency band can be extracted from the original signal. And also if we set p=p1,p2, that is to say. we define the different factor WNk×q and multiplied by the result of M-point FFT simultaneously to get multi-channel output Xp1′′,Xp2′′, and so on. As shown in Figure 2, the dashed line means that we can add another more output channels if need.Figure 2
Structure of proposed demodulation method.


## 4. Implement in HF Radar System

In high frequency (3–30 MHz) ground (surface) wave radar, electromagnetic wave propagates along the ocean surface, in the lower frequency, the radar can reach a distance of 450 kilometers [9]. The propagation of surface wave is demonstrated in Figure 3.Figure 3
HF surface wave radar (IFM, University of Hamburg).As discussed above, appropriate sampling rate and repetition time are determined. In our HF radar system,N (N = 8192) points are sampled in each sweep repetition period with sampling rate fs = 60 KHz (N=Ts/fs). During system test, a simulated “target” is generated by delaying and attenuating transmit signal, and feed it back into the receiver. The “target” (in frequency domain) appears at the 80th point, corresponding to a distance of(12)R0=80×c2α·Ts.
In the system, Ts=0.2667 s, α=112.5 kHz/s, so we have a simulated target distance of 400 km, and the offset frequency is 300 Hz (Δf=80×(1/Ts)), but the bandwidth of sampled data is fs/2  (fs/2=30 kHz). After processing with new structure, we get 512 points of the first frequency segment (M=512,Q=N/M=16), and the analytic frequency band is decreased by Q times. fs'=fs/2Q, as shown in Figure 4(b).Spectrum of processing results with proposed method.
(a)
Frequency points (1–8192)(b)
Frequency points (1–512)(c)
Frequency points (513–1024)If there is also a target at 1000th frequency point in the spectrum, the probable distance (propagating path) is nearly 3000 km. This might happen in Sky-wave Radar system, in which a certain range of distance is concerned. For example, Using once ionospheric reflection, an OTH (Over The Horizon) radar covers the range interval between about 1000 and 3500 km, as shown in Figure5. So if p is set to 1, factor WNk×q,k=[M+1,M+2,…,2M]. we can also get the second segment independently (Figure 4(c)). And the algorithm can be configured as multi-output structure, with different p1,p2,… value, we can get corresponding output Xp1′′,Xp2′′,… simultaneously.Figure 5
Demonstration of OTH Radar.The proposed algorithm is more efficient than the whole FFT calculation, the computing complexity of FFT isNlog2(N), while the complexity of proposed structure is N×(1+log2(M)) for calculating X0′′. During the calculation in DSP or FPGA, FFT calculating block and parallel technique are used, so Q channels M-point FFT can be processed simultaneously, less time is used for the processing in reality.The conventional SRC method could also derive a certain frequency band from original signal, but the shape of digital filters (anti-imaging and anti-aliasing) is not as optimal as the proposed structure. When using conventional SRC method, with higher performance of roll-off characteristics, pass-band ripples and stop-band attenuation, higher stage number of filters is demanded [10]. However, it brings more complex computing. So the proposed method provides a better “filter” performance (actually, it is not a kind of filter, in some case, has a function of filter) and less computation.


## 5. Realization of Modulation and Arbitrary Sampling Rate Conversion

The structure can be inversely used for modulation (interpolation), as shown in Figure6, but the input data is M-point frequency domain signal y[k], with the structure (Figure 6), we can get N-point N=M×Q time domain output signal y[n], But the factor need to be changed from WNk×q to WN-k×q  (q=0,1,…,Q-1;k=p×M+1,p×M+2,…,(p+1)M), and p is a value of [0,1,2,…,Q-1], and then we take IFFT to these Q channels data to get the time domain sequences, interpolate them with Q-1 zeros, and shift the interpolated data by q(q=0,1,2,…,Q-1) respectively, at last add all the interpolated sequences, so we can get x0[n](n=1,2,…,N), the sampling frequency band increases by Q times.Figure 6
Structure of the proposed interpolation method.And also we can input two or even more sequences, for example, we input two sequencesY1[k],Y2[k], and multiplied with corresponding factors WN-k1×q and WN-k2×q, (k1=p1×M+1,p1×M+1,…,(p1+1)×M;k2=p2×M+1,p2×M+1,…,(p2+1)×M), after the multiplication with factors, we can add them and send into IFFT calculating block. So the two sequences are interpolated and modulated, so with the preset p value, the input signal can be modulated by a certain pattern. Figure 7 demonstrates an interpolating example. Actually, the output result should be a time-domain sequence, but in order to explain the algorithm, we also give the spectrum of the output sequence. We set p1=1;p2=3, and after the modulation, the input signals appear in the 2nd and 4th frequency band. The spectrum of output sequence is separated into Q frequency band as shown in (10).Demonstration of proposed modulating interpolation.
(a)
Frequency points (1–512)(b)
Frequency points (1–512)(c)
Frequency points (1–8192)Combining with the decimation and interpolation methods, Arbitrary sampling rate conversion could be achieved, we definep1-demo=0,Q-demo=Q1 and p1-mo=0,Q-mo=Q2 (demo is abbr. of demodulation in Figure 2, mo is abbr of modulation in Figure 6). So the total conversion rate is Q=Q2/Q1, and also coding the frequency band unit (when certain pattern of p1-demo, p2-demo,… and p1-mo, p2-mo are predefined) could be also achieved in some communication system if it is necessary, as shown in Figure 8. All these calculations are based on the fast-calculating of M-point FFT (IFFT) block and parallel technique.Figure 8
Structure of decimation and interpolation.


## 6. Conclusion

This paper describes a demodulation method to extract the desired frequency band in FMCW Radar, which is an efficient method in some periodic transceivers like radar and sonar systems. It also provides a flexible method for the demodulation. Different frequency bands can be derived from input signal by setting correspondingp value. And we can also modulate the signal to different frequency band by inversely using the structure. All these conversion can be achieved without filters used.



---

*Source: 897429-2010-04-01.xml*