Sine-Wave Analysis

The topic of linear systems theory is primarily about linear, time-invariant (LTI) filters. A linear filter is characterized by the property that its output-signal amplitude is linearly proportional to its input-signal amplitude. A time-invariant filter, or 'constant-coefficient' filter, performs the same filtering operation at all times. Only LTI filters can be characeterized by their impulse response, or their frequency response, or their poles and zeros plus a gain, or the like. Also, only LTI filters have the superposition property for audio signals, in which the filtering of an audio mix can be obtained alternatively by filtering each channel separately and adding the filtered channels together. Equivalently, only LTI filters exhibit no intermodulation distortion for audio signals. — Click for https://ccrma.stanford.edu/~jos/filters/Linear_Time_Invariant_Digital_Filters.html

The frequency response is defined for LTI filters as the Fourier transform of the filter output signal divided by the Fourier transform of the filter input signal — Click for https://ccrma.stanford.edu/~jos/filters/Frequency_Response_I.html

The phase response of an LTI filter is the angle of the (complex) frequency response — Click for https://ccrma.stanford.edu/~jos/filters/Phase_Response_I_I.html

The amplitude response of an LTI filter is simply the magnitude of the (complex) frequency response — Click for https://ccrma.stanford.edu/~jos/filters/Amplitude_Response_I_I.html

A sinusoid is any function of the form A sin(ω t+φ), where t is the independent variable, and A, ω, φ are fixed parameters of the sinusoid called the amplitude, (radian) frequency, and phase, respectively. Sinusoidal motion is produced by any 'pure' vibration, such as that of an ideal tuning fork or mass-spring system. — Click for https://ccrma.stanford.edu/~jos/mdft/Sinusoids.html

A signal is typically a real-valued function of time. A discrete-time signal is typically a real-valued function of discrete time, and is therefore a time-ordered sequence of real numbers. — Click for http://ccrma.stanford.edu/~jos/filters/Definition_Signal.html

The sampling rate of a discrete-time signal is defined as the number of samples per second. Its units are thus in Hertz (Hz). Shannon's Sampling Theorem states that the original continuous-time signal can be recovered exactly from the samples if and only if the sampling rate is higher than twice the highest frequency present in the original signal. Any higher frequencies will alias to frequencies below half the sampling rate. — Click for https://ccrma.stanford.edu/~jos/mdft/Sampling_Theory.html

A filter in the audio signal processing context is any operation that accepts a signal as an input and produces a signal as an output. Most practical audio filters are linear and time invariant, in which case they can be characterized by their impulse response or their frequency response. — Click for https://ccrma.stanford.edu/~jos/filters/What_Filter.html

Sine-Wave Analysis

Suppose we test the filter at each frequency separately. This is called sine-wave analysis.^2.1Fig.1.6 shows an example of an input-output pair, for the filter of Eq.(1.1), at the frequency

Hz, where

denotes the sampling rate. (The continuous-time waveform has been drawn through the samples for clarity.) Figure 1.6a shows the input signal, and Fig.1.6b shows the output signal.

**Figure 1.6:** Input and output signals for the filter . (a) Input sinusoid $x(n) = A_1 \sin (2\pi f n T + \phi _1)$ at amplitude , frequency , and phase $\phi _1=0$ . (b) Output sinusoid $y(n) = A_2 \sin (2\pi f nT + \phi _2)$ at amplitude , frequency , and phase $\phi _2 = - \pi /4$ .
$\includegraphics{eps/kfig2p4}$

The ratio of the peak output amplitude to the peak input amplitude is the filter gain at this frequency. From Fig.1.6 we find that the gain is about 1.414 at the frequency

. We may also say the amplitude response is 1.414 at

The phase of the output signal minus the phase of the input signal is the phase response of the filter at this frequency. Figure 1.6 shows that the filter of Eq.(1.1) has a phase response equal to $-2\pi/8$ (minus one-eighth of a cycle) at the frequency

Continuing in this way, we can input a sinusoid at each frequency (from 0 to

Hz), examine the input and output waveforms as in Fig.1.6, and record on a graph the peak-amplitude ratio (gain) and phase shift for each frequency. The resultant pair of plots, shown in Fig.1.7, is called the frequency response. Note that Fig.1.6 specifies the middle point of each graph in Fig.1.7.

Not every black box has a frequency response, however. What good is a pair of graphs such as shown in Fig.1.7 if, for all input sinusoids, the output is 60 Hz hum? What if the output is not even a sinusoid? We will learn in Chapter 4 that the sine-wave analysis procedure for measuring frequency response is meaningful only if the filter is linear and time-invariant (LTI). Linearity means that the output due to a sum of input signals equals the sum of outputs due to each signal alone. Time-invariance means that the filter does not change over time. We will elaborate on these technical terms and their implications later. For now, just remember that LTI filters are guaranteed to produce a sinusoid in response to a sinusoid--and at the same frequency.

Sine-Wave Analysis

``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition) Copyright © 2024-09-03 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-09-03 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University