Acoustic Measurements
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Abstract
In this experiment a maximum length sequence is used in place of a δ-function to create a noise-like sequence in a CLIO measurement system. A maximum length sequence (MLS) is a pseudorandom binary sequence. To be more precise, a MLS is just a computer generated sequence of binary (1’s & 0’s) that perpetuate a random noise. One of the main applications of MLS’s is measuring the impulse response of a system.
Introduction
A CLIO measurement system is used to measure an electronic filter, perform quasi-anechoic measurements of a loud-speaker within a normal room and to assess the reverberation decay of a room. The frequency responses of high and low pass filters will be analyzed. The CLIO measurement system is used in this experiment generates a random noise in order to test the frequency response of a speaker. Also, the speaker will reproduce sounds that are produced by the CLIO system and the manner in which these sounds are regurgitated will be analyzed. Further, the reverberation of a room will be analyzed and measured. This is done by producing short packets of sound with a given interval and recording the time it takes to decay to one millionth of its power.
Theory
The impulse response is considered to be the output of a system when it is introduced to some input signal for a brief amount of time; this input being the impulse. This response can be transformed to the frequency domain of the system where it can be measured as the frequency-response of the system. In order to fully capture this response the input and output domains need to be analyzed. The input signal can be synthesized by a Dirac Delta function (δ-function). This function represents the instance of a pulse that was created in a discrete time frame while the function maintains its original area. Further, this input signal has a known spectrum so one can discard the input and only consider the output of the system in order to determine what the response is.
Moreover, consider an input signal δ(t) which is applied to a linear system then the output which is aforementioned is called the impulse response h(t). Further, let a series of δ(t) functions represent the input signal so that:
The only instance where the integrand will serve its contribution is when the argument for the delta function goes to zero, i.e, when τ = t. The argument is a simple convolution which implies that the convolution of a delta function gives the original function back (does not change the function). Now, since x(t) has been written as a weighted contribution of the δ-function the output of the system or the response can be modelled by h(t – τ) which really is weighted by x(τ). Therefore, the output, y(t) is simply the convolution of the impulse response, h(t), and the input, x(t). This can be shown in integral form as:
When a Fourier transform is applied to a convolution the result is just the Fourier transform of the two functions in the convolution multiplied. This can be represented by:
Where describe the frequency spectra of the output and input signals of the system. So now a mathematical description of the frequency response can be shown as:
H(8) =9
Now since H(10) is just a Fourier transform of h(t) it has thus far been shown that the impulse response of the system and the frequency response of the system are related via the Fourier transform. Further, h(t) can be considered as the time-domain of the system whereas H(11) can be considered at the frequency-domain response. Since these two quantities are related through the Fourier transform they both are an accurate description of the system being analyzed.
Furthermore, in this experiment the CLIO measurement system will be utilized. A binary output which contains 14 bits will act as a noise sequence which technically represents the mentioned δ-function. The manner in which the noise is produced is such that several combinations of the input signal are fed into an OR-gate which makes the output analyze these combinations and these combinations turn out to be a random combination of binary. These values are sent to the +V and –V ports of the amplifier where the signal behaves as a random noise. Since the output is cyclically related to the original random sequence of binary one can related this output to the impulse response, h(t), of the system. Moreover, the impulse response of the system can easily be Fourier transformed by the computer to give the frequency response of the system.
Now the signals in this experiment are represented by discrete-time sequences. A discrete Fourier Transform (DFT) is a specific type of Fourier transform that transforms a function into the frequency domain of the original function. Now the main reason why this experiment is represented by discrete-time sequences is because a DFT requires an input function which is discrete in nature and is finite in time. This transform is calculated by a computer and as a result a large range of values can be calculated. In the CLIO system this means 8192 data points can be achieved. This value alone represents an analysis time of 160ms which is sampled at a frequency rate of 51.2 kHz. When the DFT is applied one has 4096 discrete and complex frequency points which have a separation of 6.25 Hz.
Procedure
The procedure that was followed was outlined in the write-up provided for Experiment #25.
Discussion
Preliminary ExperimentNormalization of the gain and time delays
The minimum delay caused by the anti-aliasing filter is 23ms. Also, the frequency response that represents unity gain for the analyser setting is -5dB.
Experiment 1: Time and Frequency responses for a filter
The measured time constant for the high-pass filter is 9.375 x 10-4 s and the measured time constant for the low-pass filter is 1.26 x 10-3 s. The frequency breakpoint (critical frequency) for the high-pass filter was measured to be 725 Hz. Further, the theoretical value for the frequency break point of a high-pass filter is 740 Hz. The measured value is relatively close to the theoretical value and has a percentage difference of 2.02%. Moreover, the frequency breakpoint for the low-pass filter was measured to be 662.5 Hz. The theoretical value for the low-pass filter is also 740 Hz and these two values differ in percentage by approximately 10.5%.
Experiment 2: Frequency response of a loud speaker
The default frequency graph in comparison to the frequency response of the all time data and the truncated quasi-anechoic responses had more a defined spectrum for the frequencies that were less than 1.0 kHz. This can be accounted for the fact that for the quasi-anechoic measurement of one period of the sample was defined on the interval of 1ms and 5ms. This implies that the frequencies that have a period of larger than 4ms will not be recorded or measured; for example the echoes in the room most likely will not make it into this spectrum since the time it would take to reach the mic would be longer than the interval of 4ms. So what this essentially means is that the frequency response of the lower end of the frequency spectrum is not well defined whereas this interval of 4ms provides a better measurement of the high frequency response of the system. Therefore, in this instance the higher frequencies provide a more descriptive response of the system. Moreover, the minimum delay as already mentioned was measured to be 0.23ms. Also, the mic is placed 0.5 meters away from the speaker. In order to determine the speed of sound the first part of the impulse response graph was analyzed for a time difference. The delay time was subtracted from the time that it took for the impulse response to spike. The speed of sound was calculated to approximately be 316 m/s. Further, the loud speaker and mic were placed close to a wall in order for a reflection to occur. This reflection was evident in the impulse response graph. There were undulations present in both the frequency and impulse response graphs. These can be thought of as echo’s caused by the reflection of sound that comes off the wall. The separation of frequency is due to the fact that there is a path length difference of the original waveform and the echo that is recorded.
Experiment 3: Reverberation of a Room
A reverberation can be defined as the persistence of sound in a particular space (in this case a room) after the original input sound has been removed. The reverberation time can be thought of the time it would the sound to decay by its decibel value in this case it would be 60dB. Further, the calculated reverberation times for the various frequencies and instances (open/close door) are tabulated as below:
Case I: Open Door | Case II: Closed Door | ||
Frequency |
RT60 (ms) |
Frequency |
RT60 (ms) |
125Hz |
270.9 ± 21.2 |
125Hz |
281.1 ± 22.1 |
500Hz |
265.5 ± 19.8 |
500Hz |
249.9 ± 20.5 |
2000Hz |
270.9 ± 20.6 |
2000Hz |
277.8 ± 20.9 |
8000Hz |
276.0 ± 21.8 |
8000Hz |
293.4 ± 22.7 |
The calculation of the reverberation time was conducted between the two horizontal lines on the Shroeder plot. Based on the data it can be observed that the reverberation times (60dB) were higher for when the door was closed as opposed to when the door was open. Moreover, it can be observed from the calculated reverberation times that as the frequency is increased the reverberation time also increases. It makes sense for the 8000 Hz to have the longest reverberation time since the sound wave would have more energy and hence taking a longer time to fully decay. Moreover, the reverberation time for the 500 Hz is particularly interesting. The RT value for 500Hz is smaller than the RT for 125Hz. This can be accounted for the fact that the 500 Hz frequency produced a sound with a wavelength corresponding to the dimensions of the room; this would cause for the reflected waves to cancel out with the original waves leaving for a shorter decay time.
Conclusion
In conclusion, the minimum delay caused by the anti-aliasing filter was determined to be 23ms. The frequency response that represents unity gain for the analyser setting was also determined to be -5dB. Moreover, the measured time constant for the high-pass and low-ass filters were determined to be 9.375 x 10-4 s and 1.26 x 10-3 s respectively. The frequency breakpoints or the critical values for the high and low pass filters were relatively close to the theoretical/accepted values. These values were determined to be 725 Hz and 662.5 Hz respectively. When the frequency response of the loud speaker was analyzed it was concluded that the lower frequencies did not have a well defined spectrum. The higher frequencies during this part of the experiment were more of an accurate measure of the frequency response of the system. Lastly, the reverberation time that was calculated changed when the door of the room was either closed or left open. In the open door instance the shorter decay times were because the waves produced did not reflect within the room as much in the case of the closed door.
References
“E25 Acoustic Measurements”, University of Waterloo, Physics 360A, 2008.