![]() a)Įxplain what type of general information would be available to the researcher if this estimation of h( t) were accurate. The ion exchange process of a cell is estimated to have the following impulse response: h( t) = e −4 t u( t). 20.Ī brief current pulse of duration 50 ms and amplitude 1 mA is presented to a cell membrane with time constant 10 ms. Given x( t) = e − at u( t) and h( t) = e − bt u( t) where a and b are constants greater than zero, explain why it would be easier to evaluate the convolution x( t)* h( t) in the frequency domain. Prove the shift property of the Fourier transform. Hint: Find a few points on the curve by substituting values for the variable. ![]() 17.įind the Fourier transform of f( t) = e −3| t| and sketch its time and frequency domain representations. 15.įind the Fourier transform of f 1( t) = e −3 t u( t). 14.Įxplain why the exponential Fourier series requires negative frequencies. Write f( t) in its compact trigonometric Fourier series form. Additional harmonics add fine details but do not contribute significantly to the raw waveform.į ( t ) = 3 + cos ( t ) + 2 cos ( 2 t ) + 4 sin ( 2 t ) - 4 ( e j 4 t + e - j 4 t 2 ) The mean plus the first and second harmonics provide the basis for the general systolic and diastolic shape, since the amplitudes of these harmonics are large and contribute substantially to the reconstructed waveform. Figure 11.11 shows several levels of harmonic reconstruction. ![]() The amplitude coefficients, A m, are plotted on a log 10 scale so the smaller values are magnified and are therefore visible. Note that the low-frequency coefficients are large in amplitude, whereas the high-frequency coefficients contain little energy and do not contribute substantially to the reconstruction. Figure 11.10 plots the coefficients for the cosine series representation as a function of the harmonic number. Figures 11.10 and 11.11 illustrate a harmonic reconstruction of an aortic pressure waveform obtained by applying a Fourier series approximation. In practice, many periodic or quasi-periodic biological signals can be accurately approximated with only a few harmonic components. ![]()
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