Harmonic quadrature demodulation for extracting the

Harmonic quadrature demodulation for extracting the envelope of the 2nd harmonic component Sang-Min Kim*a, Jae-Hee Songb, Tai-Kyong Songa Dept of Electronic engineering, Sogang University, Seoul, Korea b Interdisciplinary Program in Biofusion Technology, Seoul, Korea a

Abstract An efficient method for separating the harmonic component (2f0) from the fundamental component (f0) using harmonic quadrature demodulation is presented. In the proposed method, the focused ultrasound signal is mixed with cosine and sine signal waveforms of harmonic frequency 2f0 to produce the inphase and quadrature components, respectively. The quadrature component is Hilbert-transformed and then added to the inphase component. This process cancels out both the high and low frequency components of the mixed fundamental signal and the high frequency component of the mixed harmonic signal, leaving only the envelope of the harmonic signal at the base band. This signal is then fed to a low-pass filter to remove out of band noise. In summary, this method can extract the harmonic signal after a single transmit-receive event even when there exists frequency overlap between the f0 and 2f0 components. Hence, the proposed method is superior to the pulse inversion method which requires twice as many transmit-receive cycles as well as the conventional filtering method which has a bandwidth limitation. Therefore, one can find the proposed method useful not only for tissue harmonic imaging but also for contrast agent imaging in applications where high frame rate or low motion artifact is important. The proposed method is verified by both the analytic and computer simulation studies. For a stationary target, the difference between the estimated harmonic signals by the proposed and the pulse inversion methods is within 0.1%. Keywords: harmonic quadrature demodulation, focused ultrasound signal, single transmit-receive event, frequency overlap, pulse inversion method

1. INTRODUCTION Harmonic imaging in diagnostic ultrasound imaging is the method to construct an image using the harmonic signal of the received signal and is widely used for ultrasound contrast imaging1 and tissue harmonic imaging2 due to its higher spatial/contrast resolution compared to the fundamental imaging. In harmonic imaging, it is of crucial importance to extract only the second harmonic signal from the received signal. There are two widely used approaches for separating the harmonic and the fundamental signals : one is to use a band pass filter to reject the fundamental signal2 and the other employs the pulse inversion technique3. The former method has a problem of bandwidth limitation. That is, the transmitted signal should be band-limited to avoid the overlap between the fundamental and the harmonic signals, otherwise the overlapped part of the harmonic signal should be filtered out on receive. On the other hand, the latter method allows the bandwidth overlap and yet can remove the fundamental signal completely by adding up two consecutively received signals after transmitting the ultrasound signal with opposite polarities. However, this method requires twice as many transmit-receive cycles, resulting in decrease in frame rate4, 5 and possibly motion artifact. In this work, we propose a new harmonic imaging method that can completely remove the fundamental signal without the bandwidth limitation, requiring only a single transmit-receive cycle. To be more specific, the proposed method uses a novel harmonic quadrature demodulation method devised for eliminating the fundamental component, leaving the harmonic component intact even when there exists the bandwidth overlap between the two components. Since the proposed method requires a single transmit-receive event along each scanline, it can be called fast pulse inversion technique. A method for applying the harmonic quadrature demodulation in a practical environment where the phase of the transmit signal may be altered during the propagation through the medium and in the focusing process is proposed. The proposed method is verified through the theoretical analysis and MATLAB simulations. The simulation results show that the proposed method provides almost the same result as that by the pulse inversion technique.

Medical Imaging 2008: Ultrasonic Imaging and Signal Processing, edited by Stephen A. McAleavey, Jan D'hooge, Proc. of SPIE Vol. 6920, 692017, (2008) · 1605-7422/08/$18 · doi: 10.1117/12.770040

Proc. of SPIE Vol. 6920 692017-1 2008 SPIE Digital Library -- Subscriber Archive Copy

2.

THE THEORY OF THE HARMONIC QUADRATURE DEMODULATION v1 (n)

r ( n)

v4 (n)

cos(2π 2 f 0 n ) v2 (n)

H!IPGL 1L9UOLLJJGL

bE

y ( n)

v3 (n)

sin(2π 2 f 0 n ) Fig. 1. Signal processing block diagram of the proposed harmonic quadrature demodulation method

Fig. 1 demonstrates the basic idea of the harmonic quadrature demodulation (HQD) employed in the proposed harmonic extraction method. The proposed HQD requires two multipliers, a Hilbert transformer, an adder and a LPF to extract the second harmonic signal from the focused ultrasound signal, denoted r(n). For the purpose of simple analysis without loss of generality, let’s express the focused ultrasound signal r(n) as

r (n) = a (n) cos(2π f 0 n + φ1 ) + b(n) cos(2π 2 f 0 n + φ2 ) ,

(1)

where the first term represents the envelope of the fundamental signal and the second term is the second harmonic signal. It is assumed that any DC component is removed by the transducer and an appropriate filtering operation prior to HQD. Now, the outputs of the two quadrature mixers are expressed as follows :

v1 (n) = r (n) cos(2π 2 f 0 n) = a (n) cos(2π f 0 n + φ1 ) cos(2π 2 f 0 n) + b(n) cos(2π 2 f 0 n + φ2 ) cos(2π 2 f 0 n)

(2)

1 1 1 1 = b(n) cos(φ2 ) + a (n) cos(2π f 0 n − φ1 ) + a(n) cos(2π 3 f 0 n + φ1 ) + b( n) cos(2π 4 f 0 n + φ2 ) 2 2 2 2 and

v2 (n) = r (n) sin(2π 2 f 0 n) = a (n) cos(2π f 0 n + φ1 ) sin(2π 2 f 0 n) + b(n) cos(2π 2 f 0 n + φ2 ) sin(2π 2 f 0 n)

(3)

1 1 1 1 = − b(n) sin(φ2 ) + a(n) sin(2π f 0 n − φ1 ) + a( n) sin(2π 3 f 0 n + φ1 ) + b( n) sin(2π 4 f 0 n + φ2 ) 2 2 2 2 One can see from (2) and (3) that both v1(n) and v2(n) contain the envelope of the fundamental signal located at f0 and 3f0 and the envelope of the harmonic signal located at zero frequency and 4f0.

Proc. of SPIE Vol. 6920 692017-2

By using the Hilbert transformer, v2(n) is phase shifted by -90°, yielding

v3 (n) = hilbert [ v2 (n)] (4) 1 1 1 1 = −hilbert[ b(n)sin(φ2 )] − a(n)cos(2π f0 n − φ1 ) − a(n)cos(2π 3 f0 n + φ1 ) − b(n)cos(2π 4 f0 n + φ2 ) 2 2 2 2 Finally, adding up v1(n) and v3(n) gives

1 1 v4 (n) = v1 (n) + v3 (n) = cos(φ2 )b(n) − sin(φ2 )hilbert[b(n)] 2 2

(5)

Note that v4(n) only contains baseband component, implying that the LPF in Fig. 1 is not necessary. As a matter of fact, the LPF in Fig. 1 is used only for suppressing the out-of-band noise. Let’s consider the case where Φ2 = 0. In this case, the HQD in Fig. 1 produces 0.5b(n), whether or not there exists bandwidth overlap between the fundamental and the harmonic signals. In reality, however, Φ2 is not zero, and hence the HQD would provide the distorted envelope of the harmonic signal as can be seen from (5).

r ( n)

bP 4!k

J34OL

φ2 v1 (n) v4 (n) cos(2π 2 f 0 n + φ 2 )

v2 (n)

H!IPGL 1LSUOLL JJGL

bE

y ( n)

v3 (n)

sin(2 π 2 f 0 n + φ 2 ) V\J!XGL

Fig. 2. A suggested approach for the application of the harmonic quadrature demodulation for practical harmonic imaging based on phase estimation.


To solve the above mentioned problem, a practical method to extract the second harmonic signal based on phase estimation and cancellation is also suggested. It is obvious that if the phase of the harmonic signal can be estimated correctly, the output of the modified HQD-based harmonic extraction method in Fig. 2 will be exactly y(n) = 0.5b(n).

r1 (n)

r ( n)

r5 (n)

r7 ( n)

∑

− tan −1 (

cos(2π 2 f 0 n)

+

r2 (n)

r6 ( n)

-

∑

r8 (n) ) r7 ( n)

φ2

r8 (n)

sin(2π 2 f 0 n)

r3 (n)

Hilbert tranformer

cos(2π 2 f 0 n) r4 ( n) sin(2π 2 f 0 n) Fig. 3. Suggested harmonic phase estimator for the modified HQD-based harmonic extractor shown in Fig. 2.

Fig. 3 shows the block diagram of the harmonic phase estimator suggested for this purpose. It is easy to show that r5(n) and r6(n) are given by

r5 (n) = r1 (n) + r4 (n) = b(n) cos(φ2 ) + a(n) cos(2π f 0 n − φ1 )

(6)

r6 (n) = r2 (n) − r3 (n) = −b(n) sin(φ2 ) + a(n) sin(2π f 0 n − φ1 )

(7)

Then, the two windowed integrators (or accumulators) will produce

r7 (n) =

∞

∑

m =−∞

r5 (m) w(n − m) =

n+ L

∑ ( b(m) cos(φ ) + a(m) cos(2π f m − φ ) )e

m=n− L

2

0

and


≈ cos φ2 B(0)

− jwn

1

w= 0

(8)

r8 (n) =

∞

n+ L

∑ r (m)w(n − m) = ∑ ( −b(m)sin(φ ) + a(m)sin(2π f m − φ ) ) e

m =−∞

6

2

m=n− L

0

≈ − sin φ2 B(0),

− jwn

1

w= 0

(9) where w(n) is a rectangular window of length 2L+1. Note that if the window length is properly selected, the two accumulators will produce approximately cosΦ2 B(0) and -sinΦ2 B(0). This is because each accumulator output can be regarded as its DTFT value at zero frequency. Consequently, one can get the estimate of Φ2 as follows :

⎛ r8 (n) ⎞ sin φ2 B(0) ⎞ −1 ⎛ −1 ⎟ = − tan ⎜ − ⎟ = − tan ( − tan φ2 ) ⎝ cos φ2 B(0) ⎠ ⎝ r7 (n) ⎠

φ2 = − tan −1 ⎜

(10)

3. SIMULATION RESULTS Computer simulations using MATLAB are conducted to verify proposed method and its performance. It is assumed that a ultrasound signal of a center frequency of 3Mhz and 33% 6dB bandwidth is transmitted and the received signal is composed of the transmitted signal and its squared term. The harmonic component is weighted so that its amplitude is smaller than that of the fundamental signal by about 10dB. Fig. 4 shows the simulation results when the phase Φ1 and Φ2 in (1) are both assumed to be zero, where Fig. 4 (a), (c), (e) and (g) represent the time waveforms of r(n), v1(n), v3(n) and y(n) in Fig. 1, respectively . For each of the time waveform, its frequency spectrum is plotted on its right hand side panel where the fundamental component and the harmonic component are represented by the dashed and solid lines, for convenience.

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-50 -15

-10

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(b)


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(g)

10

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(h)

Fig. 4. The simulation results of the proposed method shown in Fig 1. The time waveform of (a) the focused ultrasound signal r(n), (c) v1(n), (e) v3(n), and (g) y(n) are plotted on the left panels and their frequency spectra are plotted on their right sides in panels (b), (d), (f), and (h).

Note that the magnitude spectra of the v1(n) (in panel (c)) and v3(n) (in panel (f)) are exactly the same except for the harmonic component shifted down to the base-band (solid line around zero frequency in panel (f)). In fact, their frequency spectra have the opposite polarities as can be observed from their time waveforms in panels (c) and (e). Consequently, v1(n) + v3(n) gives the envelope of the harmonic signal. Through not demonstrated in this paper, the architecture in Fig. 1 produces zero output when only the fundamental signal is used as an input. Same simulations are performed for the cases where the focused signal r(n) has an arbitrary phase Φ2 assuming Φ1= Φ2/2 for convenience. 1

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Fig. 5. Comparison between the proposed method (solid line) and the pulse inversion method (dashed line) when the second harmonic component has a constant phase of (a) Φ2=0˚, (b) Φ2=30˚, (c) Φ2=60˚ and (d) Φ2=90˚. The results show that the proposed method provides almost the same results as the pulse inversion method.

The panels in Fig. 5 show the harmonic extraction results by the modified method shown in Fig. 2 for the cases where Φ2=0˚ (a), Φ2=30˚ (b), Φ2=60˚ (c) and Φ2=90˚ (d). For the comparison purpose, the harmonic envelopes by using the pulse inversion technique are also obtained. In each panel, the results of the proposed method and the pulse inversion method shown in Fig. 5 are plotted in solid line and dashed line, respectively. The estimated phases are 0.15˚, 29.48˚, 59.02˚ and 89.02˚ when the window length is chosen to be 237. Since the phase are estimated with only a small error, it turns out that compared to the pulse inversion technique, the proposed method provides almost the same results within 0.1% error as shown in Fig. 5. To simulate the data retained for a contrast agent bubble, we use modified Rayleigh-Plesset Equation6. It is assumed that the radius, the shell thickness, the shell viscosity and the shell shear modulas of the bubble are 2um, 4nm, 0.8 pas, 50Mpas, respectively.


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(c) Fig. 6. The simulation results using modified Rayleigh-Plesset Equation (a) the time waveform of the bubble response after bandpass filtering, (b) the magnitude spectrum of (a), (c) The results from the proposed method (solid line) and the pulse inversion method (dashed line)

The time waveform and the spectrum of the simulated bubble response are plotted in Fig. 6 (a) and 6 (b), respectively. As can be seen from Fig. 6 (b), the harmonic component is much smaller than the fundamental component, approximately by 40dB. The differences between the harmonic envelopes of the bubble response obtained by the proposed method (solid line) and the pulse inversion technique (dashed line), which is shown in Fig. 6(c), are very small. The mean squared error is only within 0.2%.Therefore, the results show that the proposed method can also be applied to harmonic imaging using contrast agent.


4. CONCLUSION A new harmonic imaging method using a novel harmonic quadrature demodulation is proposed to extract the second harmonic signal from the focused ultrasound signal after a single transmit-receive event. It is verified by both analytical and computer simulation that the proposed method is superior to the conventional filtering method in that it can completely remove the fundamental signal and extract only the harmonic signal even when there exists bandwidth overlap between the two components. The same result can be obtained by using the pulse inversion method. However, it requires twice as many transmit-receive cycles than the proposed method. For practical application of the proposed method, the proposed method has a drawback of requiring estimation of the phase of the harmonic signal which may limit the accuracy of the harmonic signal. In the preliminary results, however, it was shown that the phase can be correctly estimated by the proposed phase estimation method. Consequently, the proposed method can be used practically for both tissue harmonic imaging and ultrasound contrast imaging. More rigorous verification of the proposed harmonic imaging method with experimental data will be performed along with the performance evaluation and optimization of the phase estimation method.

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M. Averkiou, D. Roundhill, and J. Powers, “A new imaging technique based on the nonlinear

3

properties of tissues,” in IEEE Symp. on Ultrasonics, vol. 2, pp. 1561- 1566, 1997. Hwang et al, "Two Pulse Technique For Ultrasound Harmonic Imaging," US patent, 5,951,478, 1999. F . Tranquart , N. Grenier , V. Eder and L. Pourcelot , "Clinical Use of Ultrasound Tissue Harmonic Imaging,"

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Ultrasound in Med.& Biol., vol. 25, no. 6, pp. 889∼894, 1999. Peter N. Burns, David Hope Simson and Michalakis A. Averkiou, "Nonlinear Imaging," Ultrasound in Med.& Biol., vol. 26, Supplement 1, pp. S19∼S22, 2000. Lars Hoff, "Acoustic characterization of contrast agents for medical ultrasound imaging," pp. 54~81, Kluwer Academic Publishers., 2001.

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