Procedure for the simulation experiment of four Fourier analysis method

 

 

Exp-3(a) Relationship between CTFS and CTFT of periodic signal

  

Exp-3(b) Relationship between CTFS and CTFT of aperiodic signal

 

Exp-3(c) Relationship between DTFS and DTFT of periodic signal

 

Exp-3(d) Relationship between DTFS and DTFT of aperiodic signal

 

Exp-3(e) Relationship between CTFT and DTFT

 

Exp-3(f) Relation between four Fourier methods

 

 

                   

 

 

Exp-3(a) Relationship between CTFS and CTFT of periodic signal

 

•  Run the experiment by pressing "" button.

 

• The signal source generates different types of continuous-time periodic signal which can be selected by pressing  up/down arrow of the signal selector.The default selected signal is normalozed Gaussian pulse train. The period of the selected signal can be varied by "Time period" knob. scope1 displays the selected signal.

 

 

• The magnitude plots of CTFS harmonic function and the CTFT of the selected signal are shown in scope 2 and scope 3 respectively.

 

 

• Change the period of the signal and note the similarity between CTFS and CTFT plots. This demonstrates the information equivalence between the CTFS and CTFT of the periodic signal.

 

• To stop the experiment press the "" button.

 

         Note: The representation time in CTFS analysis is one period of the signal.

 

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  Exp-3(b) Relationship between CTFS and CTFT of aperiodic signal 

 

 

• Run the experiment by pressing "" button.

 

• The signal source generates different types of continuous-time aperiodic signal which can be selected by pressing  "up/down" arrow of the signal selector. The default selected signal is rectangular pulse.
  

 

 

• The selected aperiodic signal shown in scope 1 and its CTFT magnitude plot is shown in scope 4.

 

 

• The selected aperiodic signal is repeated whose periodicity can be adjust by the "Time period" knob. The periodically repeated signal is shown in scope 2. The magnitude plot of CTFS harmonic function of the periodically repeated signal is shown in the scope 3.

 

 

• The CTFT magnitude plot of the selected aperiodic signal is also shown in the scope 3 in red color. Observe that the CTFS harmonic function plot is the sampled version of the CTFT plot with samples taken at the rate of repetition (1/period of repetition).

 

 

• Increase the period of repetition observe the increased approximation of CTFS to CTFT plot.

 

• To stop the experiment press the "" button.

 

         Note: Both the plots shown in scope 3 and scope 4 are normalized for ease of comparison.

 

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   Exp-3(c) Relationship between DTFS and DTFT of periodic signal

 

• Run the experiment by pressing "" button.

 

• The signal source generates a discrete-time periodic signal. The period of the selected signal can be varied by pressing "Period" up/down knob. Scope 1 displays the signal.
  

 

 

• The magnitude plots of DTFS harmonic function and the DTFT of the signal are shown in scope 2 and scope 3, respectively.

 

 

• Change the period of the signal and note the similarity between the DTFS and DTFT plot. This demonstrates the information equivalence between the DTFS and DTFT of a periodic signal.

 

• To stop this press the "" button.

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     Exp-3(d) Relationship between DTFS and DTFT of aperiodic signal

 

• Run the experiment by pressing "" button.

 

• The signal source generates different types of discrete-time aperiodic pulses which can be selected by pressing the up/down arrow of the signal selector. Default signal selection is Gaussian pulse. 

 

• The selected aperiodic signal shown in scope 1 and its DTFT magnitude plot is shown in scope 4.

 

• The selected aperiodic signal is repeated whose periodicity can be adjust by the "Time period" knob. The periodically repeated signal is shown in scope 2. The magnitude plot of DTFS harmonic function of the periodically repeated signal is shown in scope 3.

 

• The DTFT magnitude plot of the selected aperiodic signal is also shown in the scope 3 in red colour. Observe that the DTFS harmonic function plot is the sampled version of the DTFT plot with samples taken at the rate of repetition (1/period of repetition).

 

• Increase the period of repetition observe the increased approximation of DTFS to DTFT plot.

 

• To stop the experiment press the "" button.
  

 

         Note: Both the plots shown in scope 3 and scope 4 are normalized for ease of comparison.

 

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     Exp-3(e) Relationship between CTFT and DTFT

 

• Run the experiment by pressing "" button.
  

 

 

• This experiment shows the equivalence of CTFT of an impulse sampled continuous-time signal and the DTFT of the sequence obtained by that sampling process.

 

 

• The continuous-time signal x(t) = sinc (t) and its Fourier transform are shown in scope 1 and scope 2, respectively.

 

• The discrete-time sequence x[n] = sinc (0.5n) and its Fourier transform are shown in scope 5 and scope 6, respectively.

 

• Assume d(t) is the impulse train with period Ts which is multiplied to the continuous-time signal x(t) to produce the impulse sampled signal x_d(t) = x(t).d(t)

 

• The continuous-time signal impulse sampled signal x_d(t) and its Fourier transform are shown in scope 3 and scope 4, respectively.
  

 

  

• Modify the period of the impulse train to match the discrete-time signal with the impulse sampled signal in terms of signal values, the match being indicated in the message box.
  

 

• In case of match observe the Fourier transform of the discrete-time sequence and the impulse sampled continuous-time signal and note the equivalence between the two.

 

• To stop the experiment press the "" button on the front panel.

 

       Note: The horizontal axis of the DTFT plot is the DT frequency which when multiplied with the  sampling rate (in Hz) equals the linear frequency of the horizontal axis of the CTFT plot of the impulse sampled signal.

                           

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    Exp-3(f) Relationship between four Fourier methods

 

• This experiment demonstrates  interrelationship between Fourier analysis methods: CTFS, CTFT, DTFS, DTFT.
  

 

• Run the experiment by pressing the "" push button.

 

• The signal selector enables the selection of different types of signals. By default, triangular signal is selected.

 

• Scope 1 displays the aperiodic continuous-time signal and scope 2 displays the corresponding CTFT plot (Normalized).

 

• A periodic repetition of the continuous-time aperiodic signal in scope 1 is shown in scope 3. The corresponding CTFS coefficients(normalized) are shown in scope 4.

 

• The scope 5 displays the aperiodic discrete-time signal and scope6 displays the corresponding DTFT plot(normalized).

 

• A periodic repetition of the discrete-time signal in scope 5 is shown in scope 7 and the corresponding DTFS coefficients(normalized) are shown in scope 8.

 

• To stop the experiment press the "" button.
  

 

        Observation: from these plots we can make following conclusions

 

1. In case of periodic signals (both continuous-time and discrete-time) the corresponding Fourier spectra are discrete.
   

2. In case of discrete signals (both continuous-time and discrete-time) the corresponding Fourier spectra are repetitive.

3. From observations (1) and (2) we note that "If a signal is periodic in one domain then its Fourier transform would be discrete in other domain".

 

 

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