Question

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).

Answer 1

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

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