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Only LTI filters can be subjected to frequency-domain analysis as illustrated in the preceding chapters. After studying this chapter, you should be able to classify any filter as linear or nonlinear, and time-invariant or time-varying.
The great majority of audio filters are LTI, for several reasons: First, no new spectral components are introduced by LTI filters.
Time-varying filters, on the other hand, can generate audible sideband images of the frequencies present in the input signal (when the filter changes at audio rates). Time-invariance is not overly restrictive, however, because the static analysis holds very well for filters that change slowly with time. For discrete time (digital) systems, the impulse is a 1 followed by zeros.
In continuous time, the impulse is a narrow, unit-area pulse (ideally infinitely narrow). The harmonic components generated by the nonlinearity are often called distortion products. Summing the power of all distortion products and dividing by the power of the sinusoidal component at the fundamental frequency (the only non-distortion component) yields the total harmonic distortion (THD). The THD of a device is usually expressed in decibels (dB). In general, the output signal contains many new sinusoidal components at frequencies obtained by adding and subtracting the frequencies present in the input signal.
In the simplest case of two input frequencies f1 and f2. A truly linear filter does not cause fundamental frequency given by the inverse of the period.
If the period is in units of seconds, then the fundamental frequency is in units of Hertz (cycles per second). According to Fourier theory, every periodic signal can be expressed as a sum of sinusoids at frequencies given by integer multiples of the fundamental frequency.
For integers greater than 1, these frequencies are called harmonic frequencies, and the sinusoids at harmonic frequencies are typically called harmonics. All the examples of payday loans online filters mentioned in Chapter 1 were LTI, or approximately LTI. In the following sections, linearity and time-invariance will be formally introduced, together with some elementary mathematical aspects of signals. Definition of a Signal Definition. A real discrete-time signal is defined as any time-ordered sequence of real numbers.
Similarly, a complex discrete-time signal is any time-ordered sequence of complex numbers. In this case, the time index has physical units of seconds, but it is isomorphic to the integers.
Every vector space comes with a field of scalars which we may think of as constant gain factors that can be applied to any signal in the space. The signals and gain factors (vectors and scalars) may be either real or complex, as applications may require. By definition, a vector space is closed under linear combinations. Thus, a signal mix is represented mathematically as a linear combination of vectors. However, in floating-point numerical simulations, closure is true for most practical purposes.
In digital audio signal processing applications, such number sequences usually represent sounds. For example, digital filters are used to implement graphic equalizers and other digital audio effects.
Thus, a real digital filter maps every real, discrete-time signal to a real, discrete-time signal. A complex filter, on the other hand, may produce a complex output signal even when its input signal is real. In this book, we are concerned primarily with single-input, single-output (SISO) digital filters. When both the input and output signals are vector-valued, we have what is called a multi-input, multi-out (MIMO) digital filter.
Some of the main structural features are illustrated in the following examples.