麻省理工学院公开课:信号与系统:模拟与数字信号处理> 滤波
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two particularly significant ones as I mentioned at the time are the modulation property and convolution property.
flows more or less directly from the convolution property
Both for continuous time and for discrete time the convolution property tells us that the Fourier Transform of the convolution of 2 time functions is the product of the Fourier Transforms. Now what this means in terms of linear time-invariant filters since we know that the in the time domain the output of the linear time-invariant filter is the convolution of the input and the impulse response. It says essentially then in the frequency domain that the Fourier Transform of the output is the product of the Fourier Transform of the impulse response namely the frequency response and the Fourier Transform of the input. So the output is described through that product. Now recall also that in developing the Fourier Transform, I interpreted the Fourier Transform as the complex amplitude of a decomposition of the signal in terms of a set of complex exponentials and the frequency response with the convolution property in a fact tells us what the amplitudes how to modify the amplitudes of each of those complex exponentials as they go through the system. Now this led to the notion of filtering where the basic concept was that since we can modify the amplitudes of each of the exponentials complex exponential component separately, we can for example retain some of them and totally eliminate others, and this is the basic notion of filtering.
For example I iluustrate here an ideal lowpass filter where we pass exactly frequency component in one band and reject totally frequency components in another band. The band being passed of course referred to as the pass band and the band rejected as the stop band.
The basic difference between continuous time and discrete time for these filters is that the discrete time versions are of course periodic in frequency.
we have the frequency response of the ideal lowpass filter, and shown below it is the impulse response so here is the frequency response and below the impulse response of the ideal lowpass filter
passband cutoff frequency and a stopban cutoff frequency
for the moving average filter, for given set of filter specifications, there're many more multiplications required than for recursive filter, but there are in certain context some very important compensating benefits for the moving average filter.
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