Raised Cosine Filters
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Raised Cosine Filters |
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Pulse Shaping Raised Cosine Filters exist primarily to shape pulses for use in communications systems. Excellent background information on this subject may be found in Ken Gentile's article, 0402Gentile50.pdf, published by RF Design in April, 2002.
The ideal raised cosine filter frequency response consists of unity gain at low frequencies, a raised cosine function in the middle, and total attenuation at thigh frequencies. The width of the middle frequencies are defined by the roll off factor constant Alpha, (0<Alpha<=1). In Filter Solutions, the pass band frequency is defined as the 50% signal attenuation point. The group delay must remain constant at least out to 15 to 20 dB of attenuation. When the pass band frequency of a raised cosine filter is set to half the data rate, then the impulse response Nyquist's first criteria is satisfied in that the impulse response is zero for T = NTs, where N is an integer, and T is the data period. Filter Solutions provides analog, IIR and FIR raised cosine filters. FIR are the most accurate and are best to use. However, if it is not possible to use an FIR filter, analog filters may approximate the raised cosine response. The higher the order of the filter, the greater the raised cosine approximation. High order raised cosine filters also produce longer time delays. The lower alpha values use less bandwidth, however, they also produce more ISI due to element value errors and design imperfections. |
Mathematically, the frequency response may be written as:
Raised Cosine Frequency Response
The ideal raised cosine filter frequency response is shown below:
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 Ideal Raised Cosine Impulse Response (Alpha = 0.35) |
A typical raised cosine square wave response is shown below:
Typical Raised Cosine Square Wave Response, Alpha = 0.35
An FIR Raised cosine filter may be synthesized directly from the impulse response, which is:
Raised Cosine Impulse Response
Root Raised Cosine Filter
The ideal root raised cosine filter, frequency response consists of unity gain at low frequencies, the square root of raised cosine function in the middle, and total attenuation at thigh frequencies. The width of the middle frequencies are defined by the roll off factor constant Alpha, (0<Alpha<1). In Filter Solutions, the pass band frequency is defined as the .707 half power point.
The root raised cosine filter is generally used in series pairs, so that the total filtering effect is that of a raised cosine filter. The advantage is that if the transmit side filter is stimulated by an impulse, then the receive side filter is forced to filter an input pulse shape that is identical to its own impulse response, thereby setting up a matched filter and maximizing signal to noise ratio while at the same time minimizing ISI.
Mathematically, the frequency response may be written as:
Root Raised Cosine Frequency Response
The ideal root raised cosine filter frequency response is shown below:
Ideal Root Raised Cosine Frequency Response
An FIR Raised cosine filter may be synthesized directly from the impulse response, which is:
Root Raised Cosine Impulse Response
Symmetrical Band Pass Raised and Root Raised Cosine Filters
If low frequency filtering of a data stream is necessary, arithmetically symmetrical raised and root raised cosine filters will eliminate ISI, while at the same time provide for low frequency filtering. Filter Solutions supports such filtering in its FIR filter suite. The rounding of the filter is controlled by the "Alpha" variable, just as in low pass filters. Alpha = 0 provides a theoretically perfect brick wall, and alpha = 1 provides a completely round top.
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Data Transmission Filters
Data Transmission Filters are similar to Raised Cosine filters, but are simpler to build in that they do not require a delay equalizer, and are less effective in removing ISI. The Data Transmission solution offered by Filter Solutions eliminates only postcursor ISI (ISI following the impulse response peak), or all ISI if the pass band frequency is doubled. Example impulse responses for 7th order filters are shown below.
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 7th Order Data Transmission Filter Impulse Response Alpha=0.35 |
From inspection, the Data Transmission filter only has one point of significant precursor ISI. If the pass band frequency is doubled, the precursor ISI disappears or is insignificant. The result is that Data Transmission filters are less effective in removing ISI, or require twice the bandwidth.
Filter Solutions creates Data Transmission filters by removing the delay equalizer from the Raised Cosine filter, and employing numerical methods to remove the postcursor ISI. This solution is not unique. Other solutions for Data Transmission Filters are known to exist, such as that found in The CRC Handbook of Electrical Filters. The solutions offered by Filter Solutions has the advantage of offering the user flexibility in designing for accuracy vs. bandwidth by selecting different Alpha values.
