ICSP2014 Proceedings
Broadband Signal Recording for a Baffled Circular Microphone Array Using Circular Harmonics Decomposition Min Wang, Xiaochuan Ma, Shefeng Yan and Zhigang Shang State Key Laboratory of Acoustics, Institute of Acoustics, Chinese Academy of Sciences, Beijing, China Email:
[email protected] Abstract—The relationship between the signals recorded by a circular microphone array (CMA) and the sound pressure distribution is analysed in this paper. By applying circular harmonics (CH) decomposition to the sound pressure, the analytic expression of the sound pressure distribution around a CMA baffled on an infinite-length rigid cylinder is formulated. Signal recording models for both single frequency and broadband signals are also established under unbaffled and baffled cases. Further more, an overlap processing method is employed to alleviate the distortions produced during the segment-divided processing. Computer simulations are conducted to demonstrate the broadband signals recorded by unbaffled and baffled CMAs, and the improved performance of overlap processing. Index Terms—Broadband signal, baffle, CMA, sound pressure, CH decomposition, overlap processing.
I. Introduction Microphone array signal processing has played a very important role in sensing and data acquisition systems ranging from audio engineering, radio astronomy, radar, and sonar to wireless communication, etc [1]. For most array processing algorithms, the signals recorded by each microphone are often processed as the raw data [1], [2]. So the correctness of the recorded signals, which is related tightly to the sound pressure distribution or the array response, significantly influence the performance of array processing algorithms. Thus, the analysis of the sound pressure and the recorded signals is of great necessity before carrying out those array processing algorithms. Although linear arrays are widely used in the field of array signal processing, circular arrays, which can form uniform beams along with 360 ◦ azimuthal directions because of the circular symmetry, have received more and more attention in the past few decades due to some particular advantages [3][11]. For a circular microphone array (CMA) in free sound field, or an unbaffled CMA, it’s straightforward to determine the signals recorded by different microphones just by arriving time delays. However, for a CMA mounted on a baffle, or a baffled CMA, the signals recorded by microphones differ in amplitude as well as in phase because of the scattering of the baffle. In practice, CMAs are usually placed on the surface of baffles to be fixed. What’s more, baffled CMAs have also been proved to posses several advantages over unbaffled ones. Meyer [5] presents a CMA mounted on a rigid sphere, it is shown that the broadband performance and the white noise
978-1-4799-2186-7/14/$31.00 ©2014 IEEE
gain (WNG) have been significantly improved compared with that of an unbaffled one. Parthy et al. [6] have compared the measured and theoretical performance of a broadband CMA baffled by a rigid cylinder. Teutsch [7] has investigated the modal decomposition of sound field. He shows that some periodic dips exist in the magnitude response of all harmonics for an unbaffled CMA, while this problem can be avoided by mounting the CMA into a rigid cylindrical baffle. Due to the improved performance of a baffled CMA, many related array signal processing algorithms have been investigated. However, the analysis of broadband signals recorded by a baffled CMA is seldom found in literature. In this paper, we use a novel method, called circular harmonics (CH) decomposition which is based on decomposing the sound field into a series of CH, to analyse the sound pressure distribution and obtain the analytic expression. Then signal recording models for both single frequency and broadband signals are established under unbaffled and baffled cases. Moreover, segment-divided processing method is applied since the length of the impinging signal sequence is often large or unknown. However, distortions occur inevitably between two adjacent segments during the processing. So an overlap processing method is proposed to reduce the distortions. II. Array Data Model Consider an M-microphone array with arbitrary geometry in free sound field, impinged by a desired plane wave from direction Ω0 = (ϑ0 , ϕ0 ) and multiple interference plane waves from directions Ω d = (ϑd , ϕd ) (d = 1, . . . , D). The data received by the mth microphone can be written as [12] xm (t) = s0 [t−τm (Ω0 )]+
D
sd [t−τm (Ωd )]+nm (t)
m = 1, . . . , M,
d=1
(1) D where s0 (t), {sd (t)}d=1 , and n(t) are the desired signal, the interferences and the noise, respectively. The received data expression Eq. (1), can be rewritten in vector notation as
292
x (t) = x s (t) + x i (t) + n (t).
(2)
By applying the Fourier transform to Eq. (1), the frequency-
domain data model can be written as Xm (ω) =S 0 (ω)e− jωτm (Ω0 ) + D S d (ω)e− jωτm (Ωd ) + Nm (ω)
m = 1, . . . , M.
(3)
d=1
Similarly, Eq. (3) can also be denoted in vector notation as X (ω) = a (kk 0 )S 0 (ω) +
D
a (kk d )S d (ω) + N (ω),
(4)
d=1 D denote the array response vectors, or array where {aa(kk d )}d=0 manifold vectors [1], for the desired signal and interferences. The dth response vector can be written as
a (kk d ) = [e− jkkd p 1 , e− jkkd p 2 , . . . , e− jkkd p M ]T . T
T
T
(5)
M {pp m }m=1
In Eq. (5), are the positions of array microphones, k is defined as k = ωc e(Ω) = 2πc f e(Ω) representing the wave number vector, and ee(Ω) is the unit vector of signal direction. Since the interferences share the same form with the desired signal, only the desired signal s(t) is considered in the flowing discussions (Noise is neglected for simplicity). The data models for an unbaffled array will become ⎞ ⎛ ⎜⎜⎜ s[t − τ1 (Ω)] ⎟⎟⎟ ⎜⎜⎜ s[t − τ (Ω)] ⎟⎟⎟ 2 ⎟⎟⎟ ⎜ x (t) = x s (t) = ⎜⎜⎜⎜⎜ (6) .. ⎟⎟⎟⎟ ⎜⎜⎜ . ⎟⎟⎠ ⎝ s[t − τ M (Ω)] for time domain, and ⎛ − jkkT p 1 ⎜⎜⎜ e ⎜⎜⎜ − jkkT p 2 ⎜⎜ e X (ω) = a (kk )S (ω) = ⎜⎜⎜⎜ .. ⎜⎜⎜ . ⎜⎝ T e− jkk p M
⎞ ⎟⎟⎟ ⎟⎟⎟ ⎟⎟⎟ ⎟⎟⎟ S (ω) ⎟⎟⎟ ⎟⎠
(7)
(a)
(b)
Fig. 1. Geometric models of (a) an unbaffled circular aperture and (b) a circular aperture baffled on an infinite-length rigid cylinder.
sound pressure at any point q (with azimuth angle φ) on the continuous circular aperture can be written as p(kr, φ, ϑ s , ϕ s ) = e− jkk
Tq
q
= e jkr sin ϑ s cos (φ−ϕs ) .
(8)
Note that the temporal term e − jωt has been suppressed for simplicity. Because of the circular symmetry, Eq. (8) can be represented by a spatial Fourier series expansion as [13] p(kr, φ, ϑ s, ϕ s ) =
∞
Pn (kr, ϑ s , ϕ s )e jnφ ,
(9)
n=−∞
where {Pn (kr, ϑ s , ϕ s )}∞ n=−∞ are Fourier coefficients and given by 2π 1 Pn (kr, ϑ s , ϕ s ) = p(kr, φ, ϑ s , ϕ s )e− jnφ dφ. (10) 2π 0 Plugging Eq. (8) into Eq. (10), and after a series of mathematical operations, Eq. (10) simplifies to
for frequency domain. From Eq. (6) or Eq. (7), it’s easy and direct to determine signals recorded by each microphone of an unbaffled array using the information of the microphone positions. However, for a baffled array, both of the amplitude and the phase of the received data are different between each microphone due to the scattered sound field. In other words, the array response vectors are very different from the unbaffled ones. As it is known that the array response vector is correlated with the sound pressure distribution on the array surface. So the analysis of recorded data by an array can be translated to the problem of calculating the sound pressure distribution around the surface of the array.
Pn (kr, ϑ s , ϕ s ) = jn Jn (kr sin ϑ s )e− jnϕs Cn (kr sin ϑ s )e− jnϕs ,
(11)
where Jn (·) is the nth-order Bessel function of the first kind, and C n (kr sin ϑ s ) jn Jn (kr sin ϑ s ). Physically, the above procedure based on a spatial Fourier series expansion, can be interpreted as a decomposition of a plane wave into so-called circular harmonics (CH) on a circle with radius r [2]. These CH notated by P n (kr, ϑ s , ϕ s ) in Eq. (11) are the Fourier coefficients of the series, which can be rewritten as ◦
Pn (kr, ϑ s , ϕ s ) = jn Jn (kr sin ϑ s )e− jnϕs ,
(12)
◦
III. Decomposition of the Sound Field Using Circular Harmonics A. Continuous Circular Apertures Consider an unbaffled continuous circular aperture of radius r lying on the xy-plane, as shown in Fig. 1(a), where the center of the aperture is located at the origin of coordinates. Assuming a unit magnitude plane wave impinging from (ϑ s , ϕ s ), the
where Pn (kr, ϑ s , ϕ s ) denotes the modal response of an unbaffled circular aperture of mode n. Without loss of generality, only normal wave incidence with respect to the z axis is considered in the following discussions, i.e. ϑ s = π/2 in Fig. 1(a). Thus, for the unbaffled circular aperture, C n (kr sin ϑ s ) simplifies to
293
Cn (kr) = jn Jn (kr).
(13)
−50
−100
−150 −2 10
In principle, there are infinite CH needed to represent the sound pressure. However, the number of CH must be truncated to a maximum order N when using a UCA. The highest order N of CH that can be captured by a UCA depends on the product of the largest wave number k max of the wavefield to be decomposed and the radius r of the array. As a rule of thumb [2], N ≈ kmax r (18)
0
20log10 |Cn |(dB)
20log10 |Cn |(dB)
0
n=0 n=1 n=2 n=3 −1
10
kr
0
10
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n=0 n=1 n=2 n=3
−150 −2 10
1
10
−1
10
(a)
kr
0
10
1
10
(b)
Fig. 2. Normalized modal magnitude response Cn (kr) for (a) an unbaffled circular aperture and (b) a circular aperture baffled on an infinite-length rigid cylinder.
Alternatively, for a circular aperture mounted into an infinite-length acoustically rigid cylinder, also known as a baffled circular aperture shown in Fig. 1(b), C n (kr) becomes [13]
J (kr) Hn (kr) , (14) Cn (kr) = jn Jn (kr) − n Hn (kr) where Hn (·) is the nth-order Hankel function of the first kind. So that the nth-order CH, or the nth-order modal response for a baffled circular aperture can be written as
• Jn (kr) n Hn (kr) e− jnϕs . Pn (kr, ϕ s ) = j Jn (kr) − (15) Hn (kr) Thus, according to CH expressions Eq. (12) and Eq. (15), it is easy to obtain the sound pressure distributions around both unbaffled and baffled continuous circular apertures, which can be expressed as ∞ p(kr, φ, ϕ s ) = Pn (kr, ϕ s )e jnφ =
n=−∞ ∞
(16) Cn (kr)e
− jnϕ s jnφ
e
,
n=−∞ ◦
where Pn (kr, ϕ s ) represents Pn (kr, ϕ s ) for unbaffled case or • Pn (kr, ϕ s ) for baffled case, and n j J n (kr) unbaffled, Cn (kr) = (17) J (kr) jn Jn (kr) − Hnn (kr) Hn (kr) baffled. Fig. 2(a) and Fig. 2(b) show the normalized frequencydependent modal magnitude response of the first four harmonics, for unbaffled and baffled circular apertures, respectively. It is seen that as kr increases, more and more harmonics are gaining in strength. For the unbaffled aperture, all harmonics show periodic dips, which imply that signals carrying frequency components around these dips cannot be completely identified. While this problem disappears when a cylindrical baffle is used, as shown in Fig. 2(b). What’s more, when the aperture is baffled, the magnitude response obviously lies about 6 dB above the unbaffled case. B. Circular Microphone Arrays For real-world applications, the continuous aperture needs to be sampled by a finite number of discrete points in space, i.e. microphones. In this paper, it is assumed that this sampling is performed by an M-sensor uniform circular array (UCA).
is often chosen since the amplitude of a particular Bessel function in the CH is small when the order n exceeds its argument (kr). Detailed analysis of truncation error has been presented in [14]. Following the same reason as for the Nyquist sampling theorem of one-dimensional time signals, at least M > 2N
(19)
microphones are needed to reproduce these spatial harmonics, or to avoid spatial aliasing. As shown in Fig. (2), for low frequencies, the higher the circular harmonic order is, the smaller the magnitude response is. Thus, the sound pressure at each microphone of a CMA can be approximated as p(kr, φm , ϕ s ) = =
N n=−N N
Pn (kr, ϕ s )e jnφm (20) Cn (kr)e
− jnϕ s jnφm
e
,
n=−N
where C n (kr) is referred to Eq. (17) for both unbaffled and baffled circular arrays. Note that the above equation is a truncated approximation compared to Eq. (16), since the higher orders of CH are extraordinarily small. C. Sound Pressure Distributions Assuming the plane wave mentioned above impinging from (ϑ s , ϕ s ) = (π/2, 0), the sound pressure distributions around the continuous apertures are shown in Fig. 3 for unbaffled and baffled cases. Fig. 3(a) and Fig. 3(b) respectively depict the amplitude and the phase of the sound pressure around an unbaffled circular aperture. It is shown that the amplitude remains constant, while the phase varies with the azimuth angle φ and the argument kr. Thus, an unbafffled continuous circular aperture doesn’t make any attenuations to the sound pressure caused by a plane wave, only phase differences exist due to time delays of arrival. Note that some tiny vibrations of the sound pressure amplitude occur at high frequencies, e.g. kr > 8, as a result of the truncation errors introduced above. While Fig. 3(c) and Fig. 3(d) show the amplitude and the phase of the sound pressure around a baffled circular aperture. The amplitude of the sound pressure, which ranges from about ‘0’ to ‘2’, is a function of φ and kr due to the scattering of the cylinder. As kr increases, the influence of scattering is more and more significant. Compared with the influence on pressure amplitude caused by the scatterer, the phase of the sound pressure doesn’t suffer so much except for some differences on the back of the cylinder, as shown in Fig. 3(d).
294
10
10
1
1
Consider a CW pulse as the incident plane wave, which can be defined by j2π f t e 0 |t=n/ fs 0 t < T, s(n) = s(t)|t=n/ fs = (26) 0 otherwise.
0.8 8
8
1
6
0.4 6
0.2
kr
1
kr
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4
0 4
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1
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1 0
0
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0
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(a)
1.5
2
(b) 10
10 1.8 8
0.8 8
1.6
0.6 0.4
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6
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kr
kr
1.2 1
0 4
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−0.2 −0.4
0.6 2
2
0.4
−0.6 −0.8
0.2 0
In Eq. (26), f 0 = k0 c/2π is the frequency the CW pulse. Thus, the recorded signal of the mth microphone can be expressed in analytic signal form as pm e j2π f0 t |t=n/ fs 0 t < T, xm (n) = xm (t)|t=n/ fs = (27) 0 otherwise.
−1
0
0.5
1
φ/π
1.5
0
2
0
0.5
1
φ/π
(c)
1.5
2
(d)
Fig. 3. Sound pressure distributions on both unbaffled and baffled continuous circular apertures due to a plane wave impinging from (π/2, 0) for (a) unbaffled pressure amplitude, (b) unbaffled pressure phase, (c) baffled pressure amplitude, and (d) baffled pressure phase.
IV. Signals Recorded by Circular Arrays In this section, the CW and LFM signals recorded by each microphone of M-sensor circular arrays for unbaffled and baffled cases are discussed. Based on the relationship between array response vectors and sound pressure distributions, the frequency-domain data model in Eq. (7) becomes X (ω) = p (kr, ϕ s )S (ω),
(21)
where p(kr, ϕ s ) = p(kr, φ1 , ϕ s ), p(kr, φ2 , ϕ s ), . . . , p(kr, φ M , ϕ s ) T (22) and p(kr, φm , ϕ s ) represents the sound pressure at the mth microphone, as expressed by Eq. (20). So the signal recorded by the mth microphone in frequency domain can be written as Xm (ω) = p(kr, φm , ϕ s )S (ω).
(23)
Plugging Eq.(20) into Eq. (23) yields Xm (ω) =
N
Cn (kr)e− jnϕs e jnφm S (ω),
(24)
n=−N
where C n (kr) is referred to Eq. (17) for unbaffled and baffled cases, and the maximum order N is chosen to satisfy Eq. (18). A. Single Frequency Signal: CW For a single frequency plane wave (k = k 0 ) impinging on an M-microphone circular array, the sound pressure at each microphone is a complex constant denoted as p(k 0 r, φm , ϕ s ). So the time-domain signal recorded by the mth microphone can be obtained directly by xm (t) = pm s(t), where pm denotes p(k0 r, φm , ϕ s ) for simplicity.
(25)
In practice, the signal recorded by an individual microphone is real-valued, which can be obtained by taking the real part of xm (t) in Eq. (36), i.e. Re pm e j2π f0 t |t=n/ fs 0 t < T, x˜m (n) = x˜ m (t)|t=n/ fs = 0 otherwise. (28) From Eq. (28), it is seen that the signal recorded by the m-th microphone is directly decided by the pressure p m at the position of the microphone. A more intuitive manner to indicate the influences on the recorded signals by circular arrays can be seen in Section V. B. Broadband Signal: LFM 1) Frequency-domain implementation: Unlike the case of employing CW signal as the incident plane wave, when an LFM signal impinges on an M-microphone circular array, M the sound pressure {p(kr, φ m , ϕ s )}m=1 are not complex constant values any more, but functions of frequency (or wave number k). So a frequency-domain implementation based on discrete Fourier transform (DFT) is employed to analyze the signals recorded by microphones. Firstly, the spectrum of the incident LFM plane wave is obtained by DFT. Then the frequency-domain response expression Eq. (23) is applied to process each parallel frequency component of the signal spectrum and related sound pressure. At last, an inverse discrete Fourier transform (IDFT) is operated to get the time-domain signals recorded by the Mmicrophone circular array. It’s noteworthy that this procedure is an exceptional case of subband-divided processing with each discrete frequency component representing a subband. An LFM pulse with frequency range [ f l , fu ] impinging on the circular array is defined by ⎧
fu − fl ⎪ ⎪ ⎨ e j2π fl + 2T t t |t=n/ fs 0 t < T, (29) s(n) = s(t)|t=n/ fs = ⎪ ⎪ ⎩ 0 otherwise. The frequency spectrum of the signal recorded by the mth microphone can be obtained by Eq. (23) as Xm (i) = Xm (ω)|ω=2πi/T = p(kr, φm , ϕ s )S (ω)|ω=2πi/T ,
(30)
where S (ω)| ω=2πi/T = fft{s(n)} and k ∈ [k l , ku ]. Note that the pressure distribution or response function p(kr, φ m , ϕ s ), should also be separated into some single frequency components with respect to S (ω)| ω=2πi/T . So that the multiplying is operated between each relative frequency component.
295
By making IDFT to the frequency spectrum sequence X m (i) in Eq. (30) and taking the real part, the signal recorded by the mth microphone in time domain is obtained as x˜m (n) = Re ifft p(kr, φm , ϕ s )S (ω)|ω=2πi/T . (31)
Amplitude
0.5 0 −0.5 −1 −1.5 −2
0
0.5
1
Consider the frequency range of the incoming LFM plane wave is [kl r, ku r] = [0.5, 3]. For depicting the overlap processing better, the sampled data number N s of the LFM pulse is set to 2048 instead of 512, then the time width is
2.5
2 m=1 m=2(8) m=3(7) m=4(6) m=5
1.5 1 0.5 0 −0.5 −1 −1.5 −2
0
0.5
1
1.5
2
2.5
t/ms
(b) Fig. 4. Signals recorded by each microphone of an 8-sensor UCA due to an impinging CW plane wave for (a) an unbaffled circular array and (b) a circular array baffled on an infinite-length rigid cylinder.
2
0
0
2(−2)
2(−2)
0
0
2(−2)
2(−2)
Amplitude
Amplitude
2
0 2(−2)
m=2(8) m=3(7) m=4(6) m=5
0
0
0 2(−2)
−2
m=1
2(−2)
2(−2) 0
B. Simulations with LFM
2
(a)
A. Simulations with CW Assuming the argument of the impinging CW pulse k 0 r is 1.27, the sampled data number N s of the CW pulse is 512, and the time width is T = N s / f s = 2.56ms. By employing Eq. (28), signals recorded by each microphone can be calculated directly in time domain, as shown in Fig. 4. It is shown in Fig. 4(a) that, for an unbaffled UCA, the recorded data of each microphone share the uniform amplitude with the impinging CW plane wave, and the differences only lie in phases. This is often the case assumed in most array processing algorithms. While for the baffled case as shown in Fig. 4(b), the amplitudes as well as the phases are varying from different microphones. This is because that the recorded signal of each microphone is the superposition of the incoming plane wave and the scattered wave.
1.5 t/ms
V. Simulations Signals recorded by microphones are illustrated through computer simulations in this section. Consider an 8microphone UCA as shown in fig. 1, a CW pulse and an LFM pulse with normalized unit amplitude are used as the impinging plane waves from (ϑ s , ϕ s ) = (π/2, 0), respectively.
m=1 m=2(8) m=3(7) m=4(6) m=5
1
Amplitude
2) Overlap processing method: Consider that if the length of the impinging LFM plane wave is very long or unknown, the signal arriving at each microphone should be divided into some segments to be processed due to the memory capacity of the sampling system. However, in the following simulations, it will be seen that some severe distortions occur at both sides of each segment. The presence of these distortions is mainly because that a potential frequency-domain rectangular window has been applied to the signal spectrum during the processing. So an overlap processing method is proposed to alleviate this problem. The main idea of the overlap processing method is described as follows. Assuming that the length of each divided segment is N, and the overlap rate is β. Applying the aforementioned frequency-domain implementation method to each overlapped segment respectively, then combine the segments in order according to the same overlap rate β, a time sequence with degraded distortions is obtained. Note that when combining two adjacent segments with overlap rate β, half or β/2 of the overlapped parts for the two segments are reserved. Simulations are operated to show the improvement of overlap processing in Section V.
2 1.5
0 0
2
4
6
8
10
−2
0
2
4
6
t/ms
t/ms
(a)
(b)
8
10
Fig. 5. Signals recorded by each microphone of an 8-sensor UCA due to an impinging LFM plane wave for (a) an unbaffled circular array and (b) a circular array baffled on an infinite-length rigid cylinder.
T = Ns / f s = 10.24ms. Signals recorded by each microphone can be obtained referring to Eq. (31). The recorded signals are separated into each individual amplitude ranging from -2 to 2 in the figure to avoid looking chaotic, as shown in Fig. 5. For the unbaffled case, as shown in Fig. 5(a), signals recorded by different microphones are almost same except for varying phases. It should be stressed that the amplitudes remain unitary at the mid-stable parts, while distortions occur at the beginning and the end parts. The main reason for the distortions, which has been discussed in Section IV, is that a potential rectangular window is applied during the frequencydomain processing.
296
1.5
1
1
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0.5 Amplitude
Amplitude
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could provide raw data for array processing algorithms that follow. The signals recorded are related tightly to the sound pressure distributions around the CMA. To obtain the analytic expression of sound pressure on the surface of an infinitelength rigid cylinder, a novel method called CH decomposition has been applied. The broadband signal waveforms recorded by unbaffled and baffled CMAs are demonstrated by computer simulations. It is seen that the presence of a baffle significantly influence the signal waveforms. In addition, by applying the overlap processing method, the performance of segmentdivided processing has been improved a lot.
without overlap overlap 50%
0
Acknowledgment
−0.5
This work was supported by the Excellent Young Scientist Foundation of NSFC (Grant No. 61222107).
−1
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t/ms
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(c)
(d)
References
Fig. 6. Results of overlap processing method applied to sensor 3 of an unbaffled 8-sensor UCA. (a) Segments-divided processing without overlap. (b) Overlap processing with the overlap rate β = 50%. (c) Incoming LFM wave. (d) Processing distortions for both overlap and without overlap cases.
Fig. 5(b) shows the signals recorded by a baffled UCA. Comparing with the unbaffled case, it is clear to see the differences in signal amplitudes. Specifically, signals recorded by microphones in front of the cylindrical baffle show increased amplitudes, while the microphones on the back of the baffle possess reduced amplitudes. Further more, for a specific microphone, the signal amplitude varies with time, or with frequency more precisely, instead of keeping unitary at the mid-stable parts any more. However, the phase of each individual signal seems not to change so obviously. Simulations of the overlap processing method are discussed before we close this section. Signals recorded by sensor 3 of the unbaffled UCA are taken as an example to depict the simulation results. First, a 2048-length LFM wave sequence is divided into two segments without any overlap, each segment has a data length of 1024. The processing results are shown in Fig. 6(a). It’s obvious to see some gaps in the combination sequence due to distortions at both sides of the two divided segments. Then, the original 2048-length LFM sequence is divided into three segments with the overlap rate β being 50%. The overlap processing results are shown in Fig. 8(b), the gaps between the adjacent segments disappear. Fig. 8(c) shows the incoming LFM wave, and Fig. 8(d) depicts the errors between the processing results and the original incoming LFM plane wave. It can be seen that the overlap processing results have significantly reduced distortions compared with the other case, especially at the middle combination points. Although distortions still exist at both sides of the combined sequence, they can be ignored when the signal sequence is very long.
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VI. Conclusions In conclusion, broadband signal recording models for both unbaffled and baffled CMAs have been established, which
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