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# Sampling theory Fourier theory made easy Sampling theory Notes

## : Sampling theory Fourier theory made easy Sampling theory Notes

``` 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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