There are many digital communications applications where a real signalis centered at one fourth the sample rate, or f s /4. Thiscondition makes quadrature downconversion particularly simple.
In the event that you'd like to generate an interpolated (increasedsample rate) version of the bandpass signal but maintain its f s /4 centerfrequency, there's an efficient way to do so.
Suppose we want to interpolate by a factor of two. So the outputsample rate is twice the input sample rate, f s-out = 2f s-in . In thiscase the process is: quadrature downconversion by f s -in/4, interpolationfactor of two, quadrature upconversion by f s-ou t/4, and thentake only the real part of the complex upconverted sequence.
The implementation of this scheme is shown at the top of Figure 13-36 below.
|Figure13-36. Bandpass signal interpolation scheme, and spectra.|
The sequences applied to the first multiplier in the top signal pathare the real x(n) input and the repeating mixing sequence 1,0,-1,0.That mixing sequence is the real (or in-phase) part of the complexexponential
needed for quadrature downconversion by f s /4. Likewise,the repeating mixing sequence 0,-1,0,1 applied to the first multiplierin the bottom path is the imaginary (or quadrature phase) part of thecomplex downconversion exponential
The 2 symbol means insert one zero-valued sample between each signalat the A nodes. The final subtraction to obtain y(n) is how we extractthe real part of the complex sequence at Node D. That is, we'reextracting the real part of the product of the complex signal at Node Cmultiplied by
The shaded spectra indicate true spectral components, while thewhite spectra represent spectral replications. Of course, the samelowpass filter must be used in both processing paths to maintain theproper time delay and orthogonal phase relationships.
There are several additional issues worth considering regarding thisinterpolation process. If the amplitude loss, inherent ininterpolation, of a factor of two is bothersome, we can make the finalmixing sequences 2,0,-2,0, and 0,2,0,-2 to compensate for that loss.
Because there are so many zeros in the sequences at Node B(three-fourths of the samples), we should consider those efficientpolyphase filters for the lowpass filtering.
Finally, if it's sensible in your implementation, consider replacingthe final adder with a multiplexer (because alternate samples of thesequences at Node D are zeros). In this case, the mixing sequence inthe bottom path would be changed to 0,-1,0,1.
Usedwith the permission of the publisher, Prentice Hall, this on-goingseries of articles on Embedded.com is based on copyrighted materialfrom “UnderstandingDigital Signal Processing, Second Edition” by Richard G. Lyons. Thebook can be purchased on line.
Richard Lyons is a consultingsystems engineer and lecturer with Besser Associates. As alecturer with Besser and an instructor for the University of CaliforniaSanta Cruz Extension, Lyons has delivered digitasl signal processingseminars and training course at technical conferences as well atcompanies such as Motorola, Freescale, Lockheed Martin, TexasInstruments, Conexant, Northrop Grumman, Lucent, Nokia, Qualcomm,Honeywell, National Semiconductor, General Dynamics and Infinion.