Figure 2.1: Ideal impulse response and measured (RealReverb Plugin)
2. Reverberation Introduction
Digital artificial reverberation effects can be found in software programs, instrument amplifiers, public address systems, rack-mounted effects consoles, etc. to add a spatial component to audio. This spatial component stamps a recording with a character unique to the given environment. Whether this environment is artificial or natural, it can be just as crucial as the recorded information itself. Reverberation effects can be achieved by using any combination of techniques.
2.1 Artificial Reverberation: Impulse Response Convolution
The impulse response of an acoustic environment represents how any single digital sample will behave in the acoustic space. Figure 2.1 shows a theoretical impulse response with its three main components (direct signal, early reflections and reverberation tail) as well as a measured medium room impulse response used in the Winamp “RealReverb” plugin.
Figure 2.1: Ideal impulse response and measured (RealReverb Plugin)
For digital systems, the most natural sounding artificial reverberation is achieved by convolving an anechoic (“dry”) input sequence with an impulse response. When this is done, the output of the system will sound like the input was excited and recorded in the same manner and location as the impulse response. For example, an impulse response is measured by playing a stimulus through a speaker and recorded from the rear of an auditorium. Convolving any input with this impulse response will result in an output that sounds exactly as if the input was played through the same speaker and recorded in the exact same location in the rear of the auditorium. The impulse response is the response of a system to a unit impulse [23] and a digital audio signal is a sequence of attenuated and delayed impulses. Thus, the convolution operation defines how each of these samples behaves within the system defined by the impulse response. Several convolution techniques are discussed in section 2.1.2.
2.1.1 Measuring Impulse Response
Determining the impulse response of an acoustic space has become a rich area of research with many different methods currently in widespread use. Some consider only the geometry and the materials of a space, while others use specific sound sources and recording techniques in order to capture the acoustic signature of a room.
All acoustic spaces are designed in an attempt to make every listening location possess the same ideal sound perception. However, due to the frequency dependent directional nature of sound waves, not all locations will experience the same sound perception, and none are guaranteed to be ideal. When a single monophonic impulse response is to be used for generating artificial reverberation, great care is taken in deciding where in the acoustic space the impulse response is going to be measured. It is this location that will become the virtualized listening location for the newly reverberant signal.[2] While producing an acceptable impulse response is not within the scope of this thesis, the various techniques are examined so that educated decisions regarding impulse response selection can be made.
2.1.1.1 Geometrical Acoustics
The Image-Source model is a geometrical/materials method of determining a room’s impulse response without using an acoustic stimulus and recording device. This method examines the effects of an acoustic source in a room with corresponding sources located in image rooms with reflecting boundaries. Each of the infinite image sources will produce attenuated, filtered and delayed versions of the original acoustic input. The total model effects are summed to produce the transfer function and impulse response. It is the materials of the boundaries, contents of room model, and the size of the room model that will determine the level of attenuation of the image sources, determining the reverberation characteristics.[26]
Figure 2.2 shows an example of how the Image-Source model is used. The original source produces virtual sources (indicated by x) that are attenuated due to the materials of the boundaries and the effects of each are accumulated at the listener location (indicated by o).
Figure 2.2: Image-Source model
A common reverberation measurement is the “reverberation time.” This figure was defined in 1922 by Wallace Sabine to be the time taken for a steady-state sound to be switched off and drop in intensity by 60dB. Sabine introduced a method of predicting this time for the Image-Source model that was based on wall surface area and absorption. Since his studies were done typically using the middle C note on a pipe organ, most reverberation times (RT60) were initially given for 512 Hz, but are now usually given for octave to ⅓ octave intervals across the human hearing spectrum. RT60 values can be as high as 20 seconds for large empty concert halls and as short as 200 milliseconds in the case of an automobile. [6]
A family of geometry-based models exists that center around a spherical point source in a model room emitting ray, cone or pyramid shaped sound “traces.” These shaped traces are drawn throughout the simulated room and are reflected and attenuated at boundaries according to the materials of the walls. The effects of the traces are summed at a particular location of interest in order to realize the time/frequency characteristics the room introduces.[26] Figure 2.3 depicts a stereo ray tracing example where L and R are the left and right ear components. The geometrical shape of the traces emitted from the point source has been the most developed component of this type of measuring method. Farina has developed the pyramid-tracing simulation technique for Ramsete, an indoor and outdoor acoustic simulation program.[7]
Figure 2.3: Ray-tracing model
These Geometrical Acoustics methods for determining the impulse response of a room do not provide the most useful impulse response on which to perform musical operations. Architects generally use these techniques in the planning stages of constructing the acoustic properties of a specific space, not for modeling an existing room. A far easier and more accurate method is used for capturing actual room responses.
2.1.1.2 Acoustic Measurement
In 1997 The International Standards Organization published the revised standard #3382: “Acoustics - Measurement of the reverberation time of rooms with reference to other acoustical parameters.” This standard outlines specific measurement parameters and methods for testing rooms. The specifications include various kinds of reverberation time and clarity measurements:
Because of the frequency dependence, each measurement is evaluated over the standard 10 octave bands from 31.5 Hz to 16 kHz. For each of these frequency bands a signal-to-noise ratio of 20dB – 45dB is needed for accuracy. Background noise in rooms is completely unpredictable and varies with occupancy, frequency, materials, airflow, measurement location, and acoustic stimuli. [13]
When acoustic stimuli are used to measure an impulse response, a host of outside influences are put into play that can easily corrupt the data that is gathered. When acoustic signals are used, such as swept sinusoids, pseudo-random sequences, or noise, an amplifier and speaker are used to inject the stimulus into the room and a microphone and computer are used to capture the results. Each of these additional components will affect the validity of the measurements.
Short impulsive high-energy sources are often used to excite an acoustic space for measurement purposes. Firearms such as cap guns and blank pistols as well as handclaps and balloon bursts have been used to satisfy this need. Ideally an impulse would need to provide:
Unfortunately, none of these impulsive acoustic sources provide ideal measurement properties, but do provide quick and easy computations.[13]
An impulse response measurement can be found without using an impulse-like stimuli, but rather using a deterministic input that can be used to derive the impulse response. In 1967 Richard Heyser developed a method known as Time Delay Spectrometry (TDS) which uses a swept sine wave as the acoustic stimulus. When the product of the input and the recorded response are integrated, the result is the impulse response. Unfortunately, while little computation is required, non linearities in the system can reduce the system signal-to-noise ratio.[14]
D. Giuliani describes the use of a “chirp-like” (swept sinusoid) stimulus for measuring the impulse response of a busy office. The particular application was for hands-free speech recognition for office use using a Hidden Markov Model (HMM) speech recognizer. The “chirp-like” signal is output from a loudspeaker and recorded using a microphone array placed in the office. Since the input signal is designed to have autocorrelation close to a perfect Dirac delta function, the recorded signal can be easily deconvolved by cross-correlating the recorded signal with the original input. This will give the exact impulse response of the room (plus any contributions from the speaker/microphone arrangement).[12] The use of swept sine waves for acoustic measurements do encounter a drawback in enclosed rooms. Rooms can have signature “modes” in the form of peaks and/or notches in the frequency response. This is created when standing waves set up between walls. While listening spaces are designed to minimize these effects, a swept sine wave will bring attention to the particular frequencies that are affected.
The use of a maximal length sequence (MLS) is often used because of its flat spectrum and mathematical convenience. The MLS is an apparent random sequence of 1s and 0s which, when output from a transducer will have a white overall frequency spectrum. In hardware systems, such as MLSSA, a feedback shift register of length LSR generates the MLS signal with a clocking frequency much greater than those of acoustical interest. The pattern is periodic by:
An MLS signal can also be created within software programs such as Aurora written by Dr. Angelo Farina.[17] Once the signal is generated, it is output from a loudspeaker and recorded and saved in some digital format. To get the impulse response of the room from the recorded signal, deconvolution must be performed to remove the input dependence.
Deconvolution is a method of removing the effects of a filtering process. In the frequency domain, this can be accomplished by multiplying the output spectrum by the inverse of the transfer function. In the time domain, it can be accomplished by convolving the output with the impulse response of the inverse transfer function:
Since the input was made up of only 1s and 0s the Fast Hadamard Transform (see section 2.1.3) can be used. This transform, like the Fast Fourier Transform (FFT), makes use of symmetries to reduce required calculations. Since the FHT kernel is based on +1 and –1, only unity multiplications are performed allowing the FHT to consist only of addition operations.
Although the MLS technique does provide a quick and easy computation method, it relies on the assumption that the acoustic system measured has linear behavior and is time stationary. In addition this technique requires that the input signal used is at least as long as the room response in order to capture the full decay of the reverberation tail. However, measuring the tail accurately is problematic due to the fact that this very soft portion of the decay is being measured in the presence of a high-energy wide band noise source.[14]
The industry standard loudspeaker of choice for output of any of the above measuring signals is the dodecahedron loudspeaker. This is a 12 sided, 12 equal driver arrangement that ensures maximum omnidirectionality of the output signal.
2.1.2 Convolution Techniques
Convolution is a mathematical process that is used to define the interaction of an input signal with a linear, time-invariant system. For digital artificial reverberation, the output of a reverberator is defined as the input convolved with a particular acoustic impulse response. This method produces the most accurate effect because the impulse response defines how each sample behaves in the room and the convolution applies this to every sample. This operation is plagued with computational complexity and has evolved into two principle implementations: direct linear convolution and block convolution.
2.1.2.1 Direct Linear FIR
Convolution is a filtering process that can be described by the following equation:
For the application to artificial reverberation, x(n) is the input, h(n) is the acoustic impulse response and y(n) is the reverberated signal. This process can be realized in the time domain via an FIR (Finite Impulse Response) architecture as shown in Figure 2.4.
Figure 2.4: FIR implementation of convolution
This is a stable real-time system because each input sample will produce an output sample and the impulse response is of finite length. If the input is Nx samples long and the impulse response is Nh samples long, the output will be Nx + Nh - 1 samples long. This brute force process requires Nx∙Nh multiplication and addition operations. The output of a 1 second input with a typical room impulse response (about 2 seconds) sampled at 44.1kHz would require ~3.89 billion multiply/adds.
It is apparent that linear convolution is an exhaustive process and if the impulse response is any more that a couple of hundred points, real-time processing quickly becomes impossible with common practical processors. In fact, if the impulse response is any more than 30 points, a much more efficient algorithm can be used to compute the convolution.[21]
2.1.2.2 Direct Frequency
Convolution can also by done in the frequency domain, by taking advantage of the property that convolution in the time domain is equivalent to complex multiplication in the frequency domain:
This method cannot function as a real-time system because the entire input signal must be defined prior to any processing. The entire output sequence is determined by taking the Inverse Discrete Fourier Transform of the product of the DFTs of the input and impulse response:
The input and impulse response lengths must be the same size to ensure that the correct frequency bins of the DFT are being multiplied. Thus, the input length, Nx, and impulse response length, Nh, must be made equal. To preserve the same output length, Ny, that the direct linear method produced, the input length must be padded with Nh –1 zeros, while the impulse response length must be padded with Nx – 1 zeros. This will ensure that the output of the direct linear and direct frequency convolution processes is identical.
This method requires performing two Nx + Nh –1 DFTs, Nx + Nh –1 complex multiplications, and 1 Nx + Nh –1 IDFT. If the input and impulse response are padded with enough zeros to make Nx + Nh –1 a power of 2 the Fast Fourier Transform (FFT) algorithm can be utilized. This will reduce the required computations to less than 1% of those needed for the direct linear operation.[27] Direct Frequency convolution depends on the complete definition of the input signal before processing, thus, it cannot be used in a real-time, or quasi real-time application.
2.1.2.3 Overlap and Add
This technique processes small blocks of the input data with the full impulse response in the frequency domain, and uses the FFT to drastically reduce computational requirements. The selected input blocks are non-overlapping, sequential audio samples. For direct linear convolution, the output is expected to be the length of the input (Nx) plus the length of the impulse response (Nh) minus 1. For example, consider the input signal and impulse response in Figure 2.5.
Figure 2.5: Overlap and add example (input and impulse response)
The input length is 9 and impulse response length is 3, therefore the length of the convolution should be 9 + 3 –1 = 11. By choosing the input block size of 3, several short convolution operations can be executed instead of one lengthy computation. The output blocks will overlap by 3 – 1 = 2 samples and the final output can be found by summing the overlapping samples (Figure 2.6 and Figure 2.7)
Figure 2.6: Overlap and add example (output blocks)
Figure 2.7: Overlap and add example (sum overlapping blocks)
When selecting the length of the input block, we must zero pad the input and impulse response to maintain this output length when we multiply in the frequency domain and perform the IFFT. If using radix-2 FFTs, the FFT length (NFFT) is chosen to be the next highest power of two that is greater than Nh·Nx is then determined according to:
The input block is then zero padded up to NFFT. This provides enough room for the convolution expansion.[25] Each output block overlaps the next by a length of Nh – 1 and must be summed to maintain exact congruence with linear convolution.
Even though this form of block convolution requires 2 FFTs, 1 IFFT, and NFFT multiplications, the efficiency of the FFT algorithm allows this method to outperform linear convolution. However, no output samples are calculated until the first Nx input samples are collected and processed. If a strict real-time application is needed, this method will not satisfy; however, if the application can tolerate modest I/O delay, this is a much more efficient alternative.
2.1.2.4 Overlap and Save
The Overlap and Save block convolution method is similar to the Overlap and Add method except the input blocks are selected to overlap and the output blocks are not. When the input blocks are processed using circular convolution, the first
Nh – 1 samples are discarded, and each block is concatenated to form the total output.
Consider the same example (Figure 2.8).
Figure 2.8: Overlap and save example (overlapping input blocks and impulse response)
By selecting overlapping input blocks, the need for summing the output blocks is eliminated. The circular convolution artifacts, which show up as the first two samples, can be eliminated in each of the output blocks (Figures 2.9).
Figure 2.9: Overlap and save example (output blocks)
The output blocks can then be concatenated to form the full output (Figure 2.10).
Figure 2.10: Overlap and save example (concatenated output)
This method still imposes an I/O delay, but does not have the accumulation step for the overlap sections like the Overlap and Add method.
2.1.2.5 Minimum Delay Block Convolution
William Gardner developed a block convolution algorithm that combines the real-time architecture of linear convolution with the computational efficiency of block convolution. This hybrid architecture uses a modest FIR direct convolution stage to ensure output samples will not be delayed while input samples are being collected for the block convolution operation.[10]
2.1.3 Pertinent Transforms
Although real-time audio is a time-based phenomenon analysis based on time characteristics alone often do not suffice. The most common alternative analysis is frequency-based due to the inherent pitch qualities of audio signals. This analysis is achieved using the Fourier Transform, a mathematical operation relating an analog waveform to its analog sinusoidal components:
For digital signals, the Discrete Fourier Transform (DFT) is used:
Because the transform’s kernel is a sinusoid (e-j2fn/N) the output of this transform reveals the level of each sinusoidal component within the original signal.
The Fast Fourier Transform (FFT) is a method of calculating the DFT of a signal in a more efficient way. If N is length of a digital signal, direct application of the DFT will require N2 operations. However, by exploiting the symmetries of the DFT, the FFT requires Nlog2N operations.[23] A 30 second audio signal sampled at 44.1kHz will require 1.75 trillion operations to calculate the DFT and only 26.9 million operations to calculate the FFT (0.001% of the required DFT operations). The FFT is used in calculating the early reflection portion of the hybrid reverberation.
While the Fourier domain produces a familiar audio characteristic, it is not the only useful analysis transform. A transform with a different kernel will reveal how the signal is composed of this new kernel. The Hadamard Transform uses a kernel made of only +1 and –1. Its matrix implementation restricts it to only digital signals. [17]
Figure 2.11 shows an N = 8 Hadamard kernel.
Figure 2.11: Hadamard kernel (N = 8)
2.2 Artificial Reverberation: filtering
Only a few simple reverberating building blocks make up the basic library of digital reverberator circuits. The inverse comb filter (FIR), comb filter (IIR) and all-pass filter (IIR) are the basic structures that have been combined in different ways in an attempt to imitate the effects of various rooms. Figure 2.12 shows the measured waveform and the frequency response of the dense diffuse reverberation of the Bergamo Cathedral in Italy. It is this characteristic that is the goal of the filtering techniques explores in this chapter.
Figure 2.12: Diffuse reverberation waveform and frequency response
2.2.1 Inverse Comb Filter
Filtering techniques are used to simulate convolution with an acoustic impulse response. The simplest building block of an artificial reverberator sums every incoming sample with a delayed and attenuated version. This is accomplished with a simple delay element in a feedforward architecture. This FIR reverberator is shown in Figure 2.13.
Figure 2.13: FIR reverberator
If the feedforward coefficient is less than 1, this circuit will output the input plus a delayed and attenuated version of the input. Note, however, that regardless of the value of this coefficient, the system remains stable. This topology has difference equation:
and transfer function:
and impulse response:
The frequency and impulse response of a simple FIR reverberator is shown in Figure 2.14. This filter topology is known as an “inverse comb filter” because the notches in the frequency response resemble the teeth of an upside down comb.
Figure 2.14: Inverse comb reverberator (frequency and impulse response)
1ms delay, 75% feed-forward gain
In the z-plane (shown in Figure 2.15), this filter topology has equally spaced zeros around the unit circle with a radius specified by the attenuation constant “a.” The delay time (in samples) will determine the amount of zeros spaced from 0 to 2 and these zeros will create nulls in the magnitude response of the filter causing coloration of the signal.
Figure 2.15: Inverse comb reverberator (z-plane)
This filter, as well as parallel and cascade combinations, has a finite length impulse response and can be guaranteed to be stable regardless of the magnitude of “a.” However, room impulse responses consist of very dense series of echoes that cannot be practically realized using this architecture.
2.2.2 Comb Filter
In room acoustics, sound is reflected off of all of the walls in an enclosed space. A sound wave reflecting off of a ceiling at a given angle, will also reflect off another wall, and still another, etc. This acoustic phenomenon is not accurately represented by the FIR delay structure in Figure 2.13. It can be modeled by feeding the output back through a delay element and summed with the input. This structure is realized in the block diagram in Figure 2.16.
Figure 2.16: IIR reverberator
This topology has difference equation:
and transfer function:
and impulse response:
This filter topology is known as a “comb filter” because the peaks in the frequency response resemble the teeth of an upright comb (Figure 2.17).
Figure 2.17: Comb reverberator (frequency and impulse response)
1ms delay, 75% feedback
Each output sample of this filter, y(n), consists of the sum of the input sample, x(n), and a delayed and attenuated version of the output sample D samples prior. Of course, the output D samples prior is equal to the input sample at that time plus the output sample D samples prior.
This topology has equally spaced poles around the unit circle with a radius specified by the attenuation constant “a.” These poles will create peaks in the magnitude response of the filter causing coloration of the signal. The z-plane plot is shown in Figure 2.18.
Figure 2.18: Comb reverberator (z-plane)
Stability is the chief concern with this filter type. If the coefficient “a” is greater than 1, the poles will be outside of the unit circle and the filter will overload. Audibly this will cause severe clipping, often shutting down the application using the algorithm. If the coefficient “a” is equal to 1 (poles on the unit circle), the filter will oscillate.
The heavy peaking in the frequency response of the comb filter does not provide a realistic sounding reverberation. Comb filters are also susceptible to flutter echoes. These are distinct echoes that arrive over 25ms in time without significant attenuation. The audible effect of flutter echoes is a sharp metallic sound that is heard in a non-absorbent room with parallel walls.[2]
2.2.3 All-Pass Filters
Reverberation is generally used for its time-based effect, not to change the frequency response of the input signal. Both the “inverse comb” and “comb” filters have poles and zeros in their transfer functions which alter their frequency response. To maintain the overall frequency content of the input and still take advantage of the time characteristics of the reverberation algorithm, an “all-pass” alternative structure can be used. In the z-plane (Figure 2.19), an all pass filter has poles and zeros at the same frequency (angle) and at inverse radii such that the effects of each
cancel:
Figure 2.19: All-pass reverberator (z-plane)
The all-pass filter accomplishes its unique pole/zero distribution through a feedback and feedforward path as shown in Figure 2.20.
Figure 2.20: All-pass reverberator
It is this combination that secures an overall flat frequency response (Figure 2.21).
Figure 2.21: All-pass reverberator (frequency and impulse response)
1ms delay, 50% feedback
The all-pass reverberator has difference equation:
and transfer function:
and impulse response:
While the magnitude response of the all-pass filter is overall flat, the phase response is not. This filter will pass all frequencies equally over the course of the entire input, however this does not apply to short time intervals or transient-type sounds. Just like the comb and inverse comb, the all-pass also has a distinct and recognizable timbre.[20]
2.2.4 Implementation
Neither the all-pass, comb, nor inverse comb filter described above has the echo or frequency density to properly simulate room reverberation on its own. However, they can be used as building blocks to combine and create suitable artificial reverberation.
Figures 2.22 and 2.23 show Schroeder’s combinations of comb and all-pass filters for the purposes of modeling room reverberation.[20]
Figure 2.22: Schroeder reverberator (combination 1)
Figure 2.23: Schroeder reverberator (combination 2)
The series all-pass configuration (Figure 2.23) provides a buildup of dense reflections, and maintains a flat overall frequency response. However, this combination still suffers from the same transient coloration that the single all-pass component does. The parallel combination of comb filters with two all-pass series components (Figure 2.22) allows for a more dense buildup of echoes and when each of the comb filter poles are given the same magnitude, the frequency coloration of the individual filters is masked.[18] The addition of the series all-pass filters allow for further density buildup without adding general timbre changes.
Moorer suggested a slight improvement to the Schroeder scheme by having 6 parallel comb filters, each with a single pole low-pass filter in the feedback loop (Figure 2.24).
Figure 2.24: Moorer reverberator
The purpose of this filter is to simulate the high frequency attenuation produced by air and materials absorption. Extensive experimentation found that nothing was gained by using more complicated filters in the feedback loop; the 1st order filters function well in smearing the echoes of more impulsive input sounds. The addition of a filter in the feedback loop of the all-pass filter did nothing to improve the performance of the overall structure.[20] The filter in the feedback loop will increase high frequency rolloff each iteration, which is consistent with room acoustics. By setting the cutoff frequency at 12kHz initially the effect will be subtle and will dynamically filter the high frequencies.[1] Figure 2.25 shows the impulse and frequency response of a single comb filter with 1ms delay and feedback coefficient value of 0.85 with a single pole low-pass filter (3kHz cutoff frequency) in the feedback path.
Figure 2.25: Single comb filter with LPF in feedback loop
The addition of the low-pass filter in the feedback loop will migrate the pole/zero distribution in the z-plane, increasingly collapsing down towards the origin (Figure 2.26).
Figure 2.26: Single comb filter with LPF in feedback loop (z-plane)
The parallel comb structures that Schroeder and Moorer proposed can provide dense late echo characteristics only when delay times are selected to be as close to a prime ratio as possible. Violating this will cause discrete echoes to overlap creating noticeable peaks and/or nulls in the frequency response.
For memory considerations, the large banks of parallel comb filters (Figures 2.22 and 2.24) may quickly saturate the host platform resources because each comb filter will require a separate delay line. An implementation similar to this parallel structure is the multi-tap comb filter (Figure 2.27).
Figure 2.27: Two tap comb filter
This filter can have several outputs from a single delay line. The impulse will include the sum of the impulse responses of each tap, as well as “cross-tap delays.” These delays are those created by each tap due to the outputs of the other taps. For example, if tap1 = 3 samples and tap2 = 5 samples, the impulse response will include echoes spaced 3 samples apart due to each of the tap2 outputs and echoes spaced 5 samples apart due to each of the tap1 outputs (Figure 2.28).
Figure 2.28: Multi-tap comb filter
This architecture allows one multi-tap delay structure to generate a denser reverberation tail than its multiple parallel counterpart. To maintain output stability the feedback values must be selected so that the sum of the absolute values of each tap’s feedback gain is less than 1.[23] Delay times are selected to be relatively prime to minimize the amount of overlapping contributions from delay taps, but despite the memory advantages of the multi-tap structure, the limited range of stable feedback values prevents this structure from producing natural sounding reverberation.
The inevitable overlapping will create unnatural discrete echoes in the later reverberation tail [15] and high frequency anomalies as shown in the frequency response plots of Figure 2.29.
Figure 2.29: Multi-tap comb filter (frequency response)
While the multi-tap topology utilizes many outputs and few delay lines, the feedback matrix topology uses many delay lines and single outputs. This topology is shown in Figure 2.30.
Figure 2.30: General feedback matrix topology
Extremely dense echoes are built up when the outputs of each delay line are fed back and mixed with the inputs of all of the delay lines. The outputs of the delay lines are mixed according to a coefficient matrix. [18]
2.3 Measuring Reverberation Quality
Methods have been developed that assess the quality of artificial reverberation. Acoustic impulse responses are confined to the ISO3382 measurement parameters (section 2.1.1.2). Recursive filter networks can also utilize this measurement system by implementing the structure, exciting it with an impulse and evaluating the audio output. However, these mathematically defined networks can have easily derived formulas assessing reverberation properties. The Time and Frequency Densities have been adopted as the staple quantitative measurement of these reverberation systems. Unfortunately, only the single comb filter and the network of parallel comb filters have had these quantities defined mathematically.
2.3.1 Frequency Density
The Frequency Density, or Modal Density, is defined as the number of eigenfrequencies (frequency peaks) per Hertz. For predictable topologies mathematical relationships have been developed. For example, the frequency density of a network of parallel comb filters is:
Frequency Densities greater that 0.15 peaks per Hertz are accepted for adequate reverberation.[18] For the Moorer topology of Figure 16, the parallel comb filter bank has delay times of 50ms, 56ms, 61ms, 68ms, 72ms, and 78ms. The Frequency Density is 0.385, greater than the needed amount.
Insufficient Frequency Density can be heard as a ringing of particular frequencies when excited by an impulse-like input, or a predominance of particular frequencies with a steady-state input.
2.3.2 Time Density
The Time Density, or Echo Density, is defined as the number of reflections per second. Similar to Frequency Density, a mathematical relationship has been developed for predictable known topologies. The Time Density of a network of parallel comb filters is:
Time Densities greater than 10,000 are necessary for adequate reverberation.[18] For the Moorer topology, the Time Density is 95.67, far less than the 10,000 echoes/second needed. However, this does not account for the low-pass filter in each of the comb filter feedback loops or the series all-pass filter. Both of which are known to increase time and frequency densities.
2.3.3 Energy Decay Relief
Schroeder developed a measurement technique called the Energy Decay Curve. This provided a visualization that would display how the frequency response changed over time (or how the waveform changed over different frequencies):
Jot [19] altered the EDC to form the Energy Decay Relief (EDR) which was a 3 dimensional visualization of the time and frequency characteristics of a signal. Figure 2.31 shows the Energy Decay Relief of a single comb filter.
Figure 2.31: EDR of single comb filter
The discrete line of echoes or evident along the time axis while the notches are evident along the frequency axis.
Figure 2.32 shows the Energy Decay Relief of an all-pass filter. The same line of discrete echoes are evident along the time axis, however the flat overall frequency response can be seen as well.
Figure 2.32: EDR of single all-pass filter