Question 1

1. Consider a discrete time signal given below: $$x[n] = 4\delta[n+4] - 2\delta[n+3] + 3\delta[n+2] + 6\delta[n+1] + 3\delta[n] - 2\delta[n-1] + 4\delta[n-2]$$ Evaluate the following without computing the DTFT. a. $X\left(e^{j0}\right)$\ b. $\angle X\left(e^{j\omega}\right)$\ c. $X\left(e^{j\pi}\right)$\ d. $\int_0^{2\pi} X\left(e^{j\omega}\right) d\omega$\ e. $\int_0^{2\pi} \left|X\left(e^{j\omega}\right)\right|^2 d\omega$ [1 + 3 + 2 + 2 + 2] Question with plots of the real part and imaginary part of the DTFT of x[n] and the DTFT Y(e^jω) of y[n], followed by subparts (i)–(iv) Questions 3 and 4 including a block diagram where x[n] is multiplied by α^(-n), passed through LTI system h[n], then multiplied by α^n to give y[n]