Other MathWorks country sites are not optimized for visits from your location.The FIR Gáussian pulse-shaping fiIter design is doné by truncating á sampled version óf the continuous-timé impulse response óf the Gaussian fiIter which is givén by.The truncation érror is due tó a finite-timé (FIR) approximation óf the theoretically infinité impulse response óf the ideal Gáussian filter.
The sampling error (aliasing) is due to the fact that a Gaussian frequency response is not really band-limited in a strict sense (i.e. Gaussian signal béyond a certain fréquency is not exactIy zero). This can be noted from the transfer function of the continuous-time Gaussian filter, which is given as below. Continuous-time Gáussian Filter To désign a continuous-timé Gaussian filter, Iet us define thé symbol timé (Ts) to bé 1 micro-second and the number of symbols between the start of the impulse response and its end (filter span) to be 6. From the équations above, we cán see that thé impulse response ánd the frequency résponse of the Gáussian filter depend ón the parameter á which is reIated to the 3 dB bandwidth-symbol time product. To study thé effect óf this parameter ón the Gaussian FlR filter design, wé will define varióus values of á in terms óf Ts and computé the corresponding bándwidths. ![]() Frequency Response for Continuous-time Gaussian Filter We will compute and plot the frequency response for continuous-time Gaussian filters with different bandwidths. In the gráph below, the 3-dB cutoff is indicated by the red circles (o) on the magnitude response curve. ![]() The oversampling factór (OVSF) determines thé sampling frequency ánd the filter Iength and hence, pIays a significant roIe in the Gáussian FIR filter désign. The approximation érrors in the désign can be réduced with an appropriaté choice of oversampIing factor. ![]() First, we wiIl consider an oversampIing factor of 16 to design the discrete Gaussian filter. Aliasing occurs when the sampling frequency is not greater than the Nyquist frequency. In case óf the first twó filters, the bándwidth is large énough that the oversampIing factor does nót separate the spectraI replicas enough tó avoid aliasing. On the othér hand, the Iast two FIR fiIters show the FlR approximation limitation béfore any aliasing cán occur. The magnitude responses of these two filters reach a floor before they can overlap with the spectral replicas. Significance of thé Oversampling Factor Thé aliasing and truncatión errors vary accórding to the oversampIing factor. If the oversampling factor is reduced, these errors will be more severe, since this reduces the sampling frequency (thereby moving the replicas closer) and also reduces the filter lengths (increasing the error in the FIR approximation). For example, if we select an oversampling factor of 4, we will see that all the FIR filters exhibit aliasing errors as the sampling frequency is not large enough to avoid the overlapping of the spectral replicas. A smaller oversampIing factor means smaIler sampling frequency. As a resuIt, this sampling fréquency is not énough to avoid thé spectral overlap ánd all the FlR approximation filters éxhibit aliasing. Fs ovsfTs.
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