In LabVIEW’s modulation toolkit, several different types of sinc pulses can be applied to the modulated signal to implement a pulse-shaping filter. These are the raised cosine filter, the root raised cosine filter, and the Gaussian filter. Each of these is discussed below.
Raised Cosine Filter
The raised cosine filter is one of the most common pulse-shaping filters in communications systems. In addition, it is used to minimize intersymbol interference (ISI) by attenuating the starting and ending portions of the symbol period. Because these portions are most susceptible to creating interference from multi-path distortion, the shaping characteristics of the raised cosine filter helps reduce ISI. This impulse response for this filter is given by the equation shown below:
As the equation shows, the sinc pulse is implemented in the creation of this filter. The filter rolloff parameter, alpha(α), can range between values of 0 and 1. The impulse response of the resulting filter is shown below:
Figure 7: Impulse Response of Raised Cosine Filter
Again, the sinc pulse is shaped such that ideally, the resulting channel bandwidth will be described by the equation:
Bw = Rs (1 + α)
Ideally, the frequency response (FFT) of the sinc pulse should yield a completely square response such that a specific frequency bandwidth (Bw is exactly half of the symbol rate (Rs). However, in the non-ideal programming environment, the actual frequency response differs slightly from the ideal response (due to estimation of infinity, ∞). Below, we show the non-ideal frequency response simply by taking the FFT (logarithmic) of the impulse response. This is shown below:
Figure 8: Frequency Response of Raised Cosine Filter
In this specific example, we’ve used a rolloff factor (α) of 0.5. Note that the bandwidth of the signal is concentrated in a specific frequency range. Again this is critical in a communications system because keeping channels bandlimited is necessary to prevent adjacent channel interference.
Root Raised Cosine Filter
The root raised cosine filter produces a frequency response with unity gain at low frequencies and complete at higher frequencies. It is commonly used in communications systems in pairs, where the transmitter first applies a root raised cosine filter, and then the receiver then applies a matched filter.
Mathematically, the raised cosine filter can be defined by the following equation:
In this equation, α is the rolloff factor, which determines the sharpness of the frequency response. In addition, R is the number of samples per symbol. As the equation above illustrates, the sinc pulse is used to shape the filter so that it appears with a finite frequency response. The impulse response for this filter is shown below:
Figure 9: Impulse Response of Root Raised Cosine Filter
As the graph illustrates, the impulse response resembles the sinc pulse described previously. As mentioned earlier, this should ideally have a perfectly square frequency response. We show the actual frequency response of the root raised cosine filter below:
Figure 10: Frequency Response of Root Raised Cosine Filter
Again, note from the figure above that an FFT of the impulse response is not completely ideal due to estimation of infinity. Again, we’ve used a rolloff factor (α) of 0.5. Also note that like the raised cosine filter, the bandwidth of the signal is concentrated in a specific frequency range.
The Gaussian filter is a pulse shaping technique that is typically used for frequency shift keying (FSK) and minimum shift keying (MSK) modulation. This filter is unlike the raised cosine and root raised cosine filters because it does not implement zero crossing points. The impulse response for the Gaussian filter is defined by the following equation:
Below, we show a graphical representation of the impulse response. As described above, note that there are no zero crossings for this type of filter.
Figure 11: Impulse Response of Gaussian Filter