Reference no: EM131128489

Homework Assignment

1. Show Fourier transform pair; e^j2πf₀t x (t) ←FT→ X(f - f_o) by direct evaluation of Fourier transform of e^j2πf₀t x(t) . Assuming X (f) is band limited by -100kHz to +100kHz around DC, sketch X(f - f_o) with f_o =10.7 MHz .

2. The thermal noise density is given by N_o= kT [watt / Hz] where k = 1.38 x10^-23 Boltzmann constant and T is temperature in Kelvin scale, and 0^°C =273^° K. At room temperature 27^°C = 300^° K, the noise density N_o =1.38 x10^-23 x 300 x1000 [miliwatt /Hz] is commonly expressed by dBm. 0 dBm = 10 log (1 miliwatt). Show that N_o =-174 [dBm / Hz]. Find the noise power in dBm when the noise bandwidth is 1MHz.

3. 6-PAM is shown as below; find symbol power (variance) assuming each symbol is equally likely (i.e., probability =1/6) using the definition of variance of a random variable.

4. The impulse response of a filter is shown below with g(t) ←FT→ G(f); g(t)

Find _-∞∫^+∞j |G(f )|² df and G_p = max |G(f)|. What is the noise bandwidth (double sided) of this filter?