4.1 Lab 4: fourier series and its applications  (Page 3/3)

Doppler Effect

The Doppler effect denotes the change in frequency and wavelength of a wave as perceived by an observer moving relative to the wave source. The Doppler effect can be demonstrated via time scaling of Fourier series. The observer hears the siren of an approaching emergency vehicle with different amplitudes and frequencies as compared to the original signal. As the vehicle passes by, the observer hears another amplitude and frequency. The reason for the amplitude change (increased loudness) is because of the proximity of the vehicle. The closer it is, the louder it gets. The reason for frequency (pitch) change is due to the Doppler effect. As the vehicle approaches, each successive compression of the air caused by the siren occurs a little closer than the last one, and the opposite happens when the vehicle passes by. The result is the scaling of the original signal in the time domain, which changes its frequency. When the vehicle approaches, the scaling factor is greater than 1, resulting in a higher frequency, and, when it passes by, the scaling factor is less than 1, resulting in a lower frequency. More theoretical aspects of this phenomenon are covered in reference [link] .

Define the original siren signal as $x(t)$ . When the vehicle approaches, one can describe the signal by

where $B_1(t)$ is an increasing function of time (assuming a linear increment with time) and $a$ is the scaling factor having a value greater than 1. When the vehicle passes by, one can describe the signal by

where $B_2(t)$ is a decreasing function of time (assuming a linear decrement with time) and $b$ is the scaling factor having a value less than 1.

First, generate a signal and create an upscale and a downscale version of it. Observe the Fourier series for all the signals. Set the amplitude and frequency of the original signal and the scaling factors as controls. In addition, play the sounds using the LabVIEW Play Waveform function. [link] shows a possible front panel for this type of system.

Synthesis of Electronic Music

In electronic music instruments, sound generation is implemented via synthesis. Different types of synthesis techniques such as additive synthesis, subtractive synthesis and frequency modulation (FM) synthesis are used to create audio waveforms. The simplest type of synthesis is additive synthesis, where a composite waveform is created by summing sine wave components, which is basically the inverse Fourier series operation. However, in practice, to create a music sound with rich harmonics requires adding a large number of sine waves, which makes the approach inefficient computationally. To avoid adding a large number of sine waves, modulation with addition is used. This exercise involves the design of algorithms used in the Yamaha DX7 music synthesizer, which debuted in 1983 as the first commercially available digital synthesizer.

The primary functional circuit in DX7 consists of a digital sine wave oscillator plus a digital envelope generator. Let us use additive synthesis and frequency modulation to achieve synthesis with six configurable operators. When one adds together the output of some operators, an additive synthesis occurs, and when one connects the output of one operator to the input of another operator, a modulation occurs.

In terms of block diagrams, the additive synthesis of a waveform with four operators is illustrated in [link] .

The output for the combination shown in [link] can be written as

[link] shows the FM synthesis of a waveform with two operators.

The output for the combination shown in this figure can be written as

Other than addition and frequency modulation, one can use feedback or self-modulation in DX7, which involves wrapping back and using the output of an operator to modulate the input of the same operator as shown in [link] .

The corresponding equation is

Different arrangements of operators create different algorithms. [link] displays the diagram of an algorithm.

And the output for this algorithm can be written as

With DX7, one can choose from 32 different algorithms. As one moves from algorithm No. 32 to algorithm No. 1, the harmonics complexity increases. In algorithm No. 32, all six operators are combined using additive synthesis with a self modulator generating the smallest number of harmonics. [link] shows the diagram for all 32 combinations of operators. More details on music synthesis and the Yamaha DX7 synthesizer can be found in the [link] - [link] .

Next, explore designing a system with six operators and set their amplitude and frequency as controls. By combining these operators, construct any three algorithms, one from the lower side (for example, algorithm No. 3), one from the middle side (for example, algorithm No. 17) and the final one from the upper side (for example, algorithm No. 30). Observe the output waves in the time and frequency domains (find the corresponding Fourier series).