Sampling theory Fourier theory made easy Notes | EduRev

Created by: Aprajita

: Sampling theory Fourier theory made easy Notes | EduRev

 Page 1


Sampling theory 
 
 
Fourier theory made easy   
Page 2


Sampling theory 
 
 
Fourier theory made easy   
Sampling, FFT 
and Nyquist 
Frequency 
Page 3


Sampling theory 
 
 
Fourier theory made easy   
Sampling, FFT 
and Nyquist 
Frequency 
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-8
-6
-4
-2
0
2
4
6
8
5*sin (2 p4t) 
Amplitude = 5 
Frequency = 4 Hz 
seconds 
A sine wave 
We take an 
ideal sine wave 
to discuss 
effects of 
sampling 
Page 4


Sampling theory 
 
 
Fourier theory made easy   
Sampling, FFT 
and Nyquist 
Frequency 
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-8
-6
-4
-2
0
2
4
6
8
5*sin (2 p4t) 
Amplitude = 5 
Frequency = 4 Hz 
seconds 
A sine wave 
We take an 
ideal sine wave 
to discuss 
effects of 
sampling 
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-8
-6
-4
-2
0
2
4
6
8
5*sin(2 p4t) 
Amplitude = 5 
Frequency = 4 Hz 
Sampling rate = 256 
samples/second 
seconds 
Sampling duration = 
1 second 
A sine wave signal and correct sampling  
We do sampling of 4Hz 
with 256 Hz so sampling 
is much higher rate than 
the base frequency, good 
Thus after sampling we can 
reconstruct the original signal 
Page 5


Sampling theory 
 
 
Fourier theory made easy   
Sampling, FFT 
and Nyquist 
Frequency 
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-8
-6
-4
-2
0
2
4
6
8
5*sin (2 p4t) 
Amplitude = 5 
Frequency = 4 Hz 
seconds 
A sine wave 
We take an 
ideal sine wave 
to discuss 
effects of 
sampling 
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-8
-6
-4
-2
0
2
4
6
8
5*sin(2 p4t) 
Amplitude = 5 
Frequency = 4 Hz 
Sampling rate = 256 
samples/second 
seconds 
Sampling duration = 
1 second 
A sine wave signal and correct sampling  
We do sampling of 4Hz 
with 256 Hz so sampling 
is much higher rate than 
the base frequency, good 
Thus after sampling we can 
reconstruct the original signal 
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
sin(2 p8t), SR = 8.5 Hz
An undersampled signal 
Undersampled signal can confuse you about its frequency when reconstructed. 
Because we used to small frequency of sampling. Nyquist teaches us what should 
be a good frequency 
Sampling rate 
Undersampling 
can be confusing 
Here it suggests 
a different 
frequency of 
sampled signal 
Red dots 
represent 
the sampled 
data 
Here sampling rate is 
8.5 Hz and the 
frequency is 8 Hz 
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