Answer:
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
