![]() ![]() In the case of an analog system, the practically achievable value is usually somewhat below this, due to analog filters - e.g. However if I really want to be sure about my windows I maybe should examine the frequency of another song.For example at a sampling rate of 48 kHz, frequency components up to 24 kHz can be theoretically determined. Now it is too premature to say it wouldn’t be safe to listen to this song on full volume. My windows natural frequency is right in the middle of the dominant frequencies of the song and thus may resonate due to the high volume. In the figure we can see that the most dominant frequencies occur between 10 1.5-10 2.2 Hz (30-158 Hz). As we assumed before the natural frequency of my windows are about 100 Hz. The frequency scale is plotted on log scale. Alrightīy taking a FFT result of the time signal of Kendrick Lamar’s song, we get the spectrum shown below. These are DFT’s taken on discrete time windows. To keep information about time and frequencies in one spectrum, we must make a spectrogram. By taking a FFT of a time signal, all time information is lost in return for frequency information. abs(fft) * 1 / N, width = 1.5) # 1 / N is a normalization factorĪs we can see the FFT works! It has given us information about the frequencies of the waves in the time signal.Ī FFT is a trade-off between time information and frequency information. The frequency resolution is determined by: Each discrete number output of the FFT corresponds to a certain frequency. To get a good insight in the spectrum the energy should be plotted against the frequency. We are interested in the energy of each frequency, so we can determine the absolute value of the FFT’s output. The amplitude is retrieved by taking the absolute value of the number and the phase offset is obtained by computing the angle of the number. Phase offset of a certain frequency sine wave.Amplitude of a certain frequency sine wave (energy).The complex output numbers of the FFT contains the following information: This is true for all numbers in the sequence įor real number inputs is n the complex conjugate of N - n.īecause the second half of the sequence gives us no new information we can already conclude that the half of the FFT sequence is the output we need. The numbers are each others complex conjugate. If we compare the first value of the sequence (index 0) with the last value of the sequence (index 499) we can see that the real parts of both numbers are equal and that the value of the imaginary numbers are also equal in magnitude, only one is positive and the other is negative. As we can see we get complex numbers as a result. The first two and the last two values of the FFT sequency were printed to stdout. In the above code snippet the FFT result of the two sine waves is determined. For sequences of evenly spaced values the Discrete Fourier Transform (DFT) is defined as: Step function simulated with sine wavesĪ Fourier Transform will break apart a time signal and will return information about the frequency of all sine waves needed to simulate that time signal. In the underlying figure this is illustrated, as a step function is simulated by a multitude of sine waves. This method makes use of te fact that every non-linear function can be represented as a sum of (infinite) sine waves. This is where the Fourier Transform comes in. As can clearly be seen it looks like a wave with different frequencies. The figure below shows 0,25 seconds of Kendrick’s tune. Examples of time spectra are sound waves, electricity, mechanical vibrations etc. The Fourier transform is commonly used to convert a signal in the time spectrum to a frequency spectrum. Let’s use the Fourier Transform and examine if it is safe to turn Kendrick Lamar’s song ‘Alright’ on full volume. Now say I have bought a new sound system and the natural frequency of the window in my living room is about 100 Hz. This can happen to such a degree that a structure may collapse. When the dominant frequency of a signal corresponds with the natural frequency of a structure, the occurring vibrations can get amplified due to resonance. ![]() The Fourier Transformation is applied in engineering to determine the dominant frequencies in a vibration signal. In this post I summarize the things I found interesting and the things I’ve learned about the Fourier Transform. In the last couple of weeks I have been playing with the results of the Fourier Transform and it has quite some interesting properties that initially were not clear to me. ![]()
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