Consider a signal x t t t t ( ) 5sin80 2sin160 3sin 320 = + −    , which is sampled at a frequency of 100 Hz. (a) There are three different frequency components in the signal. Determine if each of them has alias frequency. Calculate the alias frequency (or frequencies).

The frequency components are:

f_1 = 40 / pi  Hz     No Alias

f_2 = 80 / pi Hz     No Alias

f_3 = 160 / pi  Hz   Alias exist

Alias Frequency for f_3 = 49.07 Hz

Step-by-step explanation:

Given:

- The sinusoidal signal is as follows:

                        x(t) = 5*sin(80*t) + 2*sin(160*t) + 3*sin(320*t)

- The sampling frequency is F_s = 100 Hz

Find:

Determine if each of them has alias frequency.

Calculate the alias frequency  

Solution:

- The 3 components of signals are:

                           x_1(t) = 5*sin(80*t) ≡ A*sin(w_1*t)

                           x_2(t) = 2*sin(160*t) ≡ A*sin(w_2*t)

                           x_3(t) = 3*sin(320*t) ≡ A*sin(w_3*t)

- The corresponding angular speeds are:

                           w_1 = 80   2*pi*f_1

                           w_2 = 160   2*pi*f_2

                           w_3 = 320 2*pi*f_3

- The components of frequencies are:

                            f_1 = 40 / pi  Hz

                            f_2 = 80 / pi Hz

                            f_3 = 160 / pi  Hz

- The condition for aliasing is:

                            F_s < 2*f_i

Frequency f_1:

                            100 Hz > 2*40 / pi = 80 / pi   Hence, no aliasing.

Frequency f_2:

                            100 Hz > 2*80 / pi = 160 / pi   Hence, no aliasing.

Frequency f_3:

                            100 Hz < 2*160 / pi = 300 / pi   Hence, Aliasing.

We have only one frequency component which has an alias frequency i.e f_3. Now calculate the alias frequency:

                           Alias Frequency = F_s - f_i

                           Alias Frequency = F_s - f_3 = 100 - 160 / pi  Hz

                          Alias Frequency = 49.07 Hz