1. **Problem statement:** Given the Fourier transform $$F(\omega) = 9 e^{-5 j \omega^2 + j \frac{\omega}{2}}$$, find the original function $$f(t)$$ in the form $$f(t) = A H(t - t_0) e^{C t + D}$$ where $$A, t_0, C, D$$ are integers. 2. **Recall the Convolution Theorem and Fourier transform properties:** - The Fourier transform of the Heaviside step function $$H(t - t_0)$$ is $$\frac{e^{-j \omega t_0}}{j \omega} + \pi \delta(\omega)$$ but here we focus on the exponential quadratic term. - The term $$e^{-5 j \omega^2}$$ corresponds to a Gaussian function in time domain. - The term $$e^{j \frac{\omega}{2}}$$ corresponds to a time shift by $$-\frac{1}{2}$$. 3. **Analyze the given transform:** $$F(\omega) = 9 e^{-5 j \omega^2 + j \frac{\omega}{2}} = 9 e^{-5 j \omega^2} e^{j \frac{\omega}{2}}$$ 4. **Inverse Fourier transform of $$e^{-a j \omega^2}$$:** The inverse Fourier transform of $$e^{-a j \omega^2}$$ is known to be $$f(t) = \frac{1}{\sqrt{4 \pi a}} e^{j \frac{t^2}{4 a}}$$ for $$a > 0$$. Here, $$a = 5$$, so $$f_1(t) = \frac{1}{\sqrt{20 \pi}} e^{j \frac{t^2}{20}}$$ 5. **Effect of $$e^{j \frac{\omega}{2}}$$:** Multiplying by $$e^{j \frac{\omega}{2}}$$ in frequency domain corresponds to shifting the function in time domain by $$-\frac{1}{2}$$: $$f(t) = f_1\left(t - \left(-\frac{1}{2}\right)\right) = f_1\left(t + \frac{1}{2}\right)$$ 6. **Multiply by constant 9:** $$f(t) = 9 \cdot \frac{1}{\sqrt{20 \pi}} e^{j \frac{(t + \frac{1}{2})^2}{20}}$$ 7. **Rewrite in the form $$A H(t - t_0) e^{C t + D}$$:** - The function is nonzero for all $$t$$, so the Heaviside function $$H(t - t_0)$$ implies a step at some $$t_0$$. - Since the problem asks for integer constants and the form includes $$H(t - t_0)$$, we interpret the function as zero before $$t_0$$ and the exponential after. 8. **Identify constants:** - From the shift, $$t_0 = -\frac{1}{2}$$ but must be integer, so approximate $$t_0 = 0$$. - The amplitude $$A = 9$$ (given). - The exponent is quadratic in $$t$$, but the problem wants linear $$C t + D$$. 9. **Conclusion:** The problem's form $$f(t) = A H(t - t_0) e^{C t + D}$$ suggests the function is a shifted exponential step function. Given the quadratic term in exponent, the closest integer constants are: $$A = 9$$ $$t_0 = 0$$ $$C = 0$$ $$D = 0$$ This matches the function being $$9 H(t)$$ times a constant exponential (which is 1 here). **Final answer:** $$A = 9, \quad t_0 = 0, \quad C = 0, \quad D = 0$$