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Sampling Rate Conversion Overview In discretetime signal processing, it is often necessary to convert a signal from one sampling rate to another. A...

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Sampling Rate Conversion Overview In discrete time signal processing, it is often necessary to convert a signal from one sampling rate to another. A common example is the conversion from the sampling rate of a compact disk (CD); signal (44.1 KHz) to that of a Digital Audio Tape (DAT) signal (48 KHz). Another example is the aud io standard in High Definition Television (HDTV) transmission, where at least three sampling rates are supported (32, 44.1, and 48 KHz). Although in principle we may convert the signal back to analog form and resample at the desired rate, it is usually preferable to perform t he entire conversion digitally. This is clue to many considerations including the fact that conversion to analog form often introduces noise in the signal, and that digital signal processing can be much more cost effective and flexible. This MATLAB project asks you to perform a sampling rate conversion on segments of audio signals. The input audio signals are quantized to 8 bits and sampled with a sampling frequency of 11,025 Hz. You are required to convert the signal to a sampling rate of 24,000 Hz in a computationally efficient manner. Although one conceptual way of realizing this sampling rate conversion process is to upsample the signal, lowpass filter, and downsample it, a more clever implementation can lead to an implementation that is many times more efficient. To do this, you can exploit various aspects of class to optimize the system, such as multistage filter implementation, filter design, and polyphase implementation. By the end of the project, you hopefully will have a much better understanding of both the theoretical aspect of the system as well as various issues in implementing a practical DSP system at a software level. Project Goal A sampling rate converter which produces an output signal with a sampling rate which is M L times the original sampling rate can be specified as shown in Figure 1. Figure 1: Sampling rate conversion system. For an ideal sampling rate converter, the lowpass filter in Figure 1 is an ideal lowpass filter with cutoff frequency = L M c ππ ω , min The goal of this project is to design an efficient DSP algorithm that implements the system in Figure 1 subject to the following constraints: The system performs the correct sampling rate conversion from 11,025 Hz to 24,000 Hz. In particular, do not assume that 11,025 Hz 11,000 Hz. [] yn [] x n Lowpass filter L M f1997
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If the system in Figure 1 is an ideal sampling rate converter , when the input x[n] is a unit impulse ], [ n δ the output y[n] has a Fourier transform ( ) ω j e Y which corresponds to an ideal lowpass filter with a cutoff frequency at ω c = (11,025/ 24,000 ) π . For an equivalent system which you are to implement, when the input x[n] is a unit impulse ] [ n δ , the output y[n] must have a Fourier transform ( ) ω j e Y which is an approximation of a lowpass filter, and meets the specifications shown below in Table 1. Passband Cutoff ( ω p ) 000 , 24 025 , 11 π Passband Ripple ± 0.1 dB or less Stopband Frequency ( ω s ) 1.2 ω p Stopband Attenuation 70 dB or more Phase Constraints |max grpdelay min grpdelay| 720 in the passband Table 1 If your sampling rate conversion system functions properly, you should meet the specifications in Table 1, and when you play the output audio signal at 24,000 Hz, it should sound the same as the input audio signal played at 11,025 Hz. You should get a rough estimate of the efficiency of your design by determining how much computation was required to perform the sampling rate conversion on the Wagner.wav signal. The method which you will use to count the number of operations will be described shortly. You are to write a MATLAB function srconvert such that the command srconvert(in) takes the input signal in with an associated sampling rate of 11,025 Hz, and returns an output signal at a sampling rate of 24,000 Hz. (See the section on writing MATLAB functions.) Once your function srconvert.m is finalized, run the command y=srconvert([1 zeros(1,3000)]); . This will produce a vector y which contains the response of your system to a unit impulse. Then call verify(y) to verify that your design meets the design specification. The Files The project zip file contains the audio files and MATLAB functions which you will need for this project. To access the zip file, click on the link below where you found this file on the web site. In testing your system, you may find it helpful to first use a unit impulse as input to your system and then later try using real audio signals as inputs and listening to the outputs. To load a sound file into a vector x , type x=wavread( filename.wav ); . To write a sound file for the MATLAB vector x into the current directory, type wavwrite(x,sampfreq, filename.wav ) . You may test your system using any of the audio signals, although you should use the Wagner.wav signal to benchmark your system performance. f1997
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