- •Fourier transform. Spectral analysis
- •Fourier transform
- •It is possible to show that the Fourier transform is equivalent to the Laplace transform when the following condition is true:
- •Fourier series representation of a periodic signal
- •Figure 1 Example of periodic function
- •Gibbs phenomenon
- •Figure 2 The Gibbs Phenomenon for truncated Fourier series of a square wave
- •Discrete fourier transform (dft)
- •The fast fourier transform (fft). Spectrum analysis with fft and matlab
- •Standard task for laboratory work №3
- •Figure 3. Simulation model
Standard task for laboratory work №3
It is necessary to investigate the spectrum of a finite signal. Obtain the Fourier Transform of the signal. Define Magnitude (Power), Amplitude and Phase for the FFT of the signal. Attach the plot. The shape of the signal depends on individual variant (see Table1).
Table 1
-
Variants
1
2
3
4
5
6
7
8
9
Shape of a finite signal
Triangular
pulse
Rectangular
pulse
Rectangular
pulse
Triangular
pulse
Rectangular
pulse
Triangular
pulse
Rectangular
pulse
Triangular
pulse
Triangular
pulse
Signal property
height 2, centered at 5 instant time, width of 0.8
height 6, centered at 5 instant time, width of 4
height 2.5, centered at 2 instant time, width of 8,
height 5, centered at 20 instant time, width of 6, skew 0
height 1, centered at 9 instant time, width of 6
height 8, centered at 12 instant time, width of 5, skew -0.5
height 12, centered at 5 instant time, width of 8
height 4, centered at 8 instant time, width of 4
height 3, centered at 5 instant time, width of 10, skew 1
Prompt: To generate rectangular signal the following syntax is used:
rectpuls (t,w) returns a continuous, aperiodic, unity-height rectangular pulse at the sample times indicated in array t, centered about t = 0 and with a width of w, by default 1.
For triangular signal generation use command “tripuls”, namely
tripuls (T,w,s) returns a continuous, a periodic, symmetric, unity-height triangular pulse at the times indicated in array t, centered about t=0 and with a width of w (by default 1) and with skew s, where -1 < s< 1. When s is 0, a symmetric triangular pulse is generated.
. Define the Fourier Transform for periodic signal. In general, the signal has the following form
Initial data for periodic signal and its type is specified in Table 2.
Table 2
-
Variants
1
2
3
4
5
6
7
8
9
Signal’s shape
cosine
sine
cosine
sine
cosine
sine
cosine
sine
cosine
f (Hz)
10
25
40
20
10
30
20
30
40
a
1
3
0.5
2
0.5
4
1
3
2.5
Define Magnitude (Power), Amplitude and Phase for the FFT of the signal. Attach the plot.
Create in Simulink environment a model given in Figure 3 and perform simulation. The model includes a generator for sine curve, buffer, window function, FFT, vector scope and block for magnitude and phase estimation. These blocks could be easily found within Signal Processing Blockset of Simulink Library. Thus, take the generator for sine curve from Signal Processing Sources. Buffer block take from Signal Management Library. Window Function block is situated in Signal Operations Library. FFT block is situated in Transforms Library; from Math Operations Library take Complex to Magnitude-Angle Block and Vector Scope could be found in Signal Processing Sinks Library.