**1.3: Fast Fourier Transform (FFT) Fundamentals**

## 1.1: Signal Simulation

1.1

**Introduction**

Signals exist in the form electrical currents and are primarily purposed for the transmission of information and data from a point to another. Signals exits in various physicals forms among them electromagnetics, mechanical and also exist in other forms where they convert to the electrical forms for the purposes of taking measurements (Blahut, 2012). The various types and shapes of signals as well as the parameter for representation are studied n this experiment. This experiment will explain and elaborate how conversion of signals occurs from sensors into data that bears meaning and can be used for the purposes of analysis instrumentation, monitoring or even control.

**Theory**

The most appropriate sensors and transducers are used in changing a physical scenario to an electrical signal. Electrical signals occur in numerous formats such as current, voltage, resistance, inductance, frequent and capacitance. These electrical signal formats used different parameters in the representation of a signal as shown below

**Amplitude**

Amplitude is an illustration of the strength of a signal and is used in determining the maximum value of a signal.

**Frequency**

This is the measurement of the number of times of occurrence of a repeated signal with a time span of one second. Frequency is measured in /second or Hz (Kennedy, 2013).

**Phase Shift**

Phase shift defines the difference in the angles between two or more periodic signals.

This experiment uses specific signal kinds including:

- Triangular wave
- Sine wave
- Saw-tooth wave(Blahut, 2012)
- Square wave

**Experimental Procedure**

**Let the experiment run****Click on Experiment 1: Introduction to Signal Processing and pick on Experiment 1.3: FFT Fundamentals****Switch between the various types of signals using the drop-down for signal type. Take note of the changes in the shape and extract a fast draft for every signal in the appropriate box.**

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Figure 4 • Triangular Wave as signal Simulation

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Differences were observed between the various types of the waves as shown in figure 3, 4, 5 and 6 with no changes in the frequency and the amplitude in all the four different waves.

**Alter the amplitude and the frequency of the signal (A) and take note of the effect on the signals**

## 1.2: Signal Aliasing

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Variations in the amplitude between the two signal waves had an effect as illustrated in signal B (5) and signal A (2)

**Vary the phase signal (B) and take note of the phase shift**

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The variation in the phase signal has an effect on the phase shift as can be observed in the two signal waves when the phase signal B is changed from 0 to 70.

**Select on another type of signal and subject it to various characteristics**

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**Results and Discussion**

It is observed from this experiment that a change in the wave type brings about the same change between the two different signals

**1.2: Signal Aliasing**

**Introduction**

Aliasing refers to the impact resulting from an imperfect reconstruction of a signal from the original signal (Blahut, 2012). This experiment aims at illustrating the effects and the causes of signal aliasing in different methods.

**Theory**

The Sampling Theorem states that the rate of the sample must be more than twice the highest frequency inside a signal for a baseband signal to be reproduced accurately in the sampled form, mathematically illustrated as fs>2BW. An adequate sampled signal, as well as an under-sampled signal, is shown in the figure below (Kennedy, 2013).

**Experimental Procedure**

**Let the experiment run****Click on Experiment 1: Introduction to Signal Processing and then pick on Experiment 1.2: Signal Aliasing.****Choose on Sine Wave from the drop-down menu of the Signal Type. Ensure the Signal Frequency is set at 450 Hz and the Sampling Rate so to 1000 Hz**

**Is possible to extract the correct measurements from these two settings? Explain.**

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The experiment achieved the right baseband since the sampling rate was 1000Hz which was more than twice the Signal Frequency.

3**.2 Change the Signal Frequency upwards slowly and take note of the deviations in the shape of the waveform. Note the frequency at which the correct measurement is lost**

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The signal is observed to disappear at a Frequency of 500 Hz since the sample rate is 1000 Hz. At this point, it will not be possible to achieve the accurate signal for the baseband.

**Adjust the frequency until it is more than 500 Hz and take note of the following:****There are only two samples per period of the sampled waveform which is almost flat in appearance**

## 1.3: Fast Fourier Transform (FFT) Fundamentals

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**The signal is the Frequency Spectrum begins to move backward as the frequency is increased.**

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**.Fix the Signal Frequency to just less than 1500 Hz. Observe that the sampled waveform repeats itself just 1/3 times as frequent as the original. It is impossible to differentiate between the aliased components and the real components with aliasing. It is for this reason that you must ensure that the Nyquist theorem is not violated in order to make the correct measurements. How best can this issue be solved?**

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**The use of analog filtering can be used in overcoming the above challenge. This is performing using a low pass filter that eliminates the unwanted higher frequency components and is usually called an Anti-Aliasing Filter.**

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Figure 23 Signal Frequency at 1400 Hz

Setting the rate of sampling higher in relation to the measured frequency can be another alternative.

**Results and Discussion**

From the experiment, it was observed that the correct reading of the frequency could only be achieved if the sample rate was more than twice the highest frequency that is present in the signal fs>2BW (Kennedy, 2013).

**1.3: Fast Fourier Transform (FFT) Fundamentals**

**Introduction**

This experiment aims at exploring and gaining an in-depth comprehension of the basics of Fast Fourier Transform and spectrum analysis. Fast Fourier Transform is a mathematical tool that resolves a specific signal into the addition of the sines and the cosines.

**Theory**

** The Fourier Transform (FT):** This is an analog tool primarily deployed in the analysis of the contents of frequency for continuous signals

## 1.4: Signal Averaging

** The Fast Fourier Transform (FFT)** is an algorithm applied in the rapid computation of the DFT (Blahut, 2012).

** The Discrete Fourier Transform (FFT)** is a digital technique for the analysis of the contents of the frequency of discrete signals.

Below are some of the important examples

**Experimental Procedure**

**Let the experiment run****Click on Experiment 1: Introduction to Signal Processing and the select on Experiment 1.3: FFT Fundamentals.****Pick on the Signal Type of Sine Wave and adjust the Frequency. Take note of the behavior of the frequency graphs**

**Alter the amplitude and the Phase values. Are there are effects on the graphs? If Yes, how?**

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The graphs were affected as the Amplitude changed to 7.5 Hz

**Alter the Signal Type to Square Wave. How many peaks are achieved? Explain.**

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There are 5 peaks which are different from the sine waves as a result of the shape of the wave

**Alter the Duty Cycle of the Square Wave. What do you observe at 40%, 80%, and 100%?**

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Figure 32 Square Wave as signal Simulation Duty Cycle: 100%

A decrease in the number of peaks until they all disappear is observed. This pattern is attributed to the increase in the duty cycle percentage.

**Results and Discussion**

There are numerous various FFT algorithms depending on a vast range of published theories. The FFT algorithms range from complex number arithmetic which is simple in nature to group theory as well as number theory that is used in the resolution of a signal into the sum of sines and cosines (Blahut, 2012).

**1.4: Signal Averaging**

**Introduction**

This experiment aims at exploring and providing insights into the use of Averaging in the improvement of the accuracy of measurement for signals that are noisy and are rapidly changing. Averaging modes are as follows:

which means no averaging is applicable to a measurement**No Averaging**calculates the weighted mean of the sum of the squared values**RMS Averaging**makes use of complex FFT spectrum in which the real part is averaged in separate from the imaginary part (Blahut, 2012)**Vector Averaging**is done at the frequency line of each individual and maintains the RMS peak levels of the quantities whose averages have been determined from a single FFT record to the next record.**Peak Hold Averaging**

**Theory**

Averaging is important in the reducing of the errors in measurements as well as improving the frequency of the domain analysis that permits taking readings of the signals in an easy way.

**Experimental Procedure**

**Let the experiment run****Click on Experiment 1: Introduction to Signal Processing and then select on Experiment 1.4: Signal Averaging****Observe the Averaging Mode and ensure that the default averaging mode, “No Averaging” is set which would mean there is no averaging applied to the signal.**

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**Adjust the Averaging Mode to “RMS Averaging”. take note of the impact of this change to the signal fluctuations but not the noisy floor.**

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## 1.5: Signal Windowing

**Adjust the Averaging Mode to “Vector Averaging” and take note of the quantity of noise levels reduction in the synchronous signals. This change is called Coherent Averaging or Time Synchronous Averaging.**

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**Lastly, adjust the Averaging Mode to “Peak Hold”. This is an averaging mode done at the line of each individual frequent and maintains the RMS peak.**

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**Results and Discussion**

Measurement errors are reduced with signals at different averaging modes and the frequency domain analysis enhances the reading of the signals in a simple way as can be observed from the experiment (Kennedy, 2013).

**1.5: Signal Windowing**

**Introduction**

Windowing is a standard technique used in signal processing which mainly serves to reduce the frequency smearing levels that are present in a dataset of a non-periodic waveform. This experiment used numerous types of windows aimed at gaining a better comprehension of windowing.

**Theory**

Windowing forces signals to be continuous thereby generating better results by lowering the levels of smearing present in a signal.

**Experimental Procedure**

**Let the experiment run****Click on Experiment 1: Introduction to Signal Processing, and then select on Experiment 1.5: Signal Windowing.****Observe the input signal and take note of how it begins at a different point other than zer**

**Take note of the discontinuity of the input signals when the setting is at the periodic form****Click on the drop-down menu for the "Window Function" and select from the available window functions. Observe. Select on the Hanning window**

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**Blackman-Harris**

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**Observe the windowed signal and determine if it is periodic. What are the disadvantages associated with this periodicity?**

The windowed signal is windowed as observed with the continuous signal having the same amplitude. The amplitude was observed to be decreasing from the original signal. Still, the peak was also observed to be reducing.

**Results and Discussion**

From the experiment, it was observed that windowing forces signals to be continuous and thus leading to the generation of better results as well as reducing the levels of smearing presented in the signal.

**Conclusion**

The above experiments elaborate and illustrate the single processing through 5 sub-experiments. Various types and shapes of signals as well as the most critical parameter for identification of a signal are not only introduced but also explained in details in these experiments. The experiments acted as a source of immense assistance to the senior student to understanding the methods of signal processing and the primaries meant for the analysis of signals that have the contents of their signals changing rapidly over time.

**References:**

Blahut, R. E. (2012). * Algebraic Methods for Signal Processing and Communications Coding.* London: Springer Science & Business Media.

Kennedy, R. A. (2013). * Hilbert Space Methods in Signal Processing.* Cambridge: Cambridge University Press.

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