Pulse shaping and bandlimiting square waves with Root Raised Cosine filters





Here we see how the bandwidth of pulses are limited. I used this before in the psk31 transmitter but then just adjusted parameters for lowest IMD on a psk31 receiver w/o fully understanding what it was doing. This experiment clearly shows bandlimiting by adjusting the symbol rate as a function of the sample rate. We will use a vector source generated square wave with slowly varying duty-cycle from nearly DC to nearly 0, run it thru a rrc filter and watch it on the fft display with different symbol rates.

The code: 
#!/usr/bin/env python
#
# 

from gnuradio import gr
from gnuradio import eng_notation
from gnuradio.eng_option import eng_option
from gnuradio.wxgui import stdgui, fftsink
from optparse import OptionParser
import wx, random, math

class app_flow_graph (stdgui.gui_flow_graph):
    def __init__(self, frame, panel, vbox, argv):
        stdgui.gui_flow_graph.__init__ (self, frame, panel, vbox, argv)

        self.frame = frame
        self.panel = panel
        
        
        sample_rate = 50e3
	symbol_rate_div = 8

	vec = []
	for i in range(1,255):
	  for j in range(0,40):
             for k in range(0,i):
               vec+=[0]
	     for k in range(i,256):
               vec+=[100]
	for i in range(255,1,-1):
	  for j in range(0,40):
	    for k in range(0,i):
	      vec+=[0]
	    for k in range(i,256):
	      vec+=[100]


	pulse = gr.vector_source_c(vec,1)
	rrcf_coeffs = gr.firdes.root_raised_cosine(1.0, sample_rate, sample_rate/symbol_rate_div, .5, 337)
        rrcf = gr.fir_filter_ccf(1,rrcf_coeffs)

        fft, fft_win = fftsink.make_fft_sink_c (self, panel, "", 256, sample_rate)

        self.connect (pulse, rrcf)
	self.connect (rrcf, fft)
        vbox.Add (fft_win, 1, wx.EXPAND)

        

def main ():
    app = stdgui.stdapp (app_flow_graph, "USRP FFT")
    app.MainLoop ()

if __name__ == '__main__':
    main ()




Results:

Using symbol_rate_div = 1 (symbol rate equals sample rate) and a sample rate of 50Khz we see the signal uses all available bandwidth, -25 to +25Khz:



If we change the symbol rate to half the sample rate, we get this:



One forth the sample rate:



Time domain plot of the above:



One sixth the sample rate:



lastly one eighth:



and the time domain signal of symbol rate = 1/8th sample rate: