Find the maximum value of d such that the tumor could still be detected by receiving the 5-MHz signal.

We examine how to use nonlinearities to improve depth of penetration in ultrasound. The idea is to emit a pulse at 2.5 MHz and measure the returning echoes at the second harmonic, 5.0 MHz. Our imaging system has a dynamic range of L = 80 dB, and the tissue has an amplitude attenuation factor of μa = af dB cm−1 (for  dB MHz−1 cm−1).

(a) Assume a tumor is located at depth d. We emit a f0 = 2.5 MHz pulse with amplitude A0. What is its amplitude A1 at depth d? Answer in terms of A0 and d.

(b) Through nonlinear interactions in the tumor, the scattered waveform is transformed into a sawtooth wave with the same amplitude A1. It can be expressed as

What is the Fourier transform of this waveform? Your answer will involve an infinite sum.

(c) What is the amplitude A A2 of the sinusoidal component at f A1 = 5 MHz at the site of the tumor? What is its amplitude A A3 once it has traveled back to the transducer (assume a plane wave)? Answer in terms of A A0 and d.

(d) Find the maximum value of d such that the tumor could still be detected by receiving the 5-MHz signal.

(e) Consider a traditional ultrasound system operating at f A1 = 5 MHz. Calculate its depth of penetration.

(f) Design a convolution filter, h(t), which will remove all but the 5-MHz component of the signal g(t). It should eliminate frequencies outside the range of 4–6 MHz, and have unit gain for frequencies within that interval.

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