ABSTRACT

ABSTRACT:

Filters are devices used to remove unwanted parts of an input signal. There is branch of music called Computer Music that uses digital filters to process and manipulate sound while eliminating the undesired parts of the sound. In the 1990s there were premises that digital filters have a negative affect on the attack/rise and decay parts of a Sound Envelope when filtering in the time domain. In this paper methods that can be used to eliminate or reduce filtering affect on the sound envelope will be discussed.

INTRODUCTION:

A filter is a device that blocks certain things while allowing some other things to pass through. When eliminating undesired parts of an input, whether that input is an electronic signal or a liquid such as the case of engine oil, filters, which behave like a mathematical function as shown in Figure 1, come in handy. A car’s filter stops waste, abrasive metal particles, and any other hazardous substances from circulating through the engine while allowing only pure oil to circulate. In a network, “a filter is a device that impedes the passages of signals whose frequencies fall within a band called the stopband, wile permitting those in another band, the passband, to pass from input to output relatively unchanged”[Network Analysis]. Almost entirely every electronic device that we use in our daily life, such as television, radio, and telephone, has filters that are in one form or another.

unfiltered signal -> FILTER -> filtered signal

Figure 1

Filters as mentioned above are devices that eliminate undesired parts of a signal while allowing the desired parts of the signal to go through The four major filter types are Lowpass, Highpass, Bandpass, and Bandstop as illustrated in Figures 2a-2d. A lowpass filter allows all the frequencies ω that are less than or equal to some cutoff frequency ω_c to pass through, where ω_c is some arbitrarily chosen frequency. A highpass filter, on the other hand, allows all the frequencies that are higher than the cutoff frequency to go through, and again, ω_c is some arbitrarily chosen frequency. A bandpass filter has passband ω_l ≤ ω_c < ω_u, where ω_u and ω_l are upper and lower half-power frequencies. A bandstop has a split passband, all ω ≤ ω_l and all ω ≥ ω_u[Network Analysis].

a) Lowpass Filter

b) Highpass Filter

c) Bandpass Filter

d) Bandstop Filter

Figure 2: Major Filter Types

In this information age computers run everything in our daily life. An insurance company uses computers to keep a database of it customers, and a doctor uses computers to safe lives. It is computer’s existence and more user-friendly that everything has to adapt to it. A branch of music called Computer Music had emerged because of computer availability. The use of computers in audio systems can vary from simply storing a sound, manipulating that sound, or using a computer’s keyboard and mouse with some software to generate a real sound.

In the real world any sound that we hear whether it is coming from a speaker, a singer, or a musical instrument is in an analog form. Figure 3, shows a graphical representation of a sinusoidal change in air pressure versus time [Digital Audio]. Sound is also in the form of an acoustical wave, which propagates through air creating an air pressure that pushes around molecules in the air. When these air molecules strike a microphone in the same pattern as they propagated from their source, the microphone in turn produces an electrical pressure or electrical signal. This electrical pressure is called voltage and its shape is similar to their waveform of the sound [Computer Music; page 12]. However, we want to represent this electrical pressure, which will henceforth be referred to as a voltage, in a way that a computer can manipulate it or simply store it.

Figure 3: Sinusoidal change in air pressure caused by a simple vibration back and forth [Digital Audio]

When a sound is received in the form of a continues electrical signal, which is called an analog signal, we have to represent it in a digital form. There is a device called analog-to-digital converter, which exactly does the job. The process of converting an analog signal to a digital signal is called sampling and quantization method as we studied in class and in my earlier Electrical Engineering and Computer Science courses. Sampling of the signal is collecting equal discrete parts of the analog signal, and quantization is assigning a number to each amplitude of the sampled signal, as demonstrated in Figure 4. A binary representation of the quantized amplitude can now be stored in computer memory for later usage [digital audio]. Now that the signal has been converted to something that a computer can process we can manipulate the sound in a form that we desire.

Figure 4: A 4,000 Hz cosine wave under sampled at 6,000 Hz [Digital Audio]

Additive Synthesis and Amplitude Modulations are just two of the many types of sound manipulation that we can do using a computer. (Explain each, add examples, graphs).

Digital filters play an important in this digital signal processing of sound, whether this sound is produced by a computer or by a regular musical instrument like a violin, or a trumpet. As we sample and quantize an analog signal to convert it to a digital signal, we also use a digital filter. We use either a highpass r a lowpass filter, depending on whether we want to allow frequencies that are higher or lower than some arbitrarily chosen frequency (cutoff frequency) to go through. Digital filters can also be used to eliminate white noise from a sound as to make a sound continues or simply remove undesired parts of a signal. Nonetheless, there is an affect that digital filters have on the sound itself. While filtering the time-domain, a filter alters the attack/rise and decay area of the sound envelope. An envelope of the sound is “the shape of the amplitude variation during the course of a tone”, and it is divided four segments [Computer Music]. These segments of the sound envelope are attack or rise, decay, steady-state or sustain, and release as shown in Figure 5 (will be inserted later).

Figure 6: Filtering affect of a Sound Envelope [Computer Music]

APPROACH:

Mathematical Solution

y(t) = x(t)*c(t)

y(t) is the output function

x(t) is the modulating signal

c(t) is the carrier

x(t) à Ä à y(t)

↑

c(t)

x(t) ↔ X(s) represent the input signal in its Laplace Transform

H(s) impulse function