DSP Tricks: Interpolating a bandpass signal

Richard G. Lyons

July 25, 2008

Richard G. Lyons

There are many digital communications applications where a real signal is centered at one fourth the sample rate, or f_s/4. This condition makes quadrature downconversion particularly simple.

In the event that you'd like to generate an interpolated (increased sample rate) version of the bandpass signal but maintain its f_s/4 center frequency, there's an efficient way to do so.

Suppose we want to interpolate by a factor of two. So the output sample rate is twice the input sample rate, f_s-out = 2f_s-in. In this case the process is: quadrature downconversion by f_s-in/4, interpolation factor of two, quadrature upconversion by f_s-out/4, and then take only the real part of the complex upconverted sequence.

The implementation of this scheme is shown at the top of Figure 13-36 below.

Figure 13-36. Bandpass signal interpolation scheme, and spectra.

The sequences applied to the first multiplier in the top signal path are 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 complex exponential

needed for quadrature downconversion by f_s/4. Likewise, the repeating mixing sequence 0,-1,0,1 applied to the first multiplier in the bottom path is the imaginary (or quadrature phase) part of the complex downconversion exponential

The 2 symbol means insert one zero-valued sample between each signal at the A nodes. The final subtraction to obtain y(n) is how we extract the real part of the complex sequence at Node D. That is, we're extracting the real part of the product of the complex signal at Node C multiplied by
 
The spectra at various nodes of this process are shown at the bottom of Figure 13-36 above.

The shaded spectra indicate true spectral components, while the white spectra represent spectral replications. Of course, the same lowpass filter must be used in both processing paths to maintain the proper time delay and orthogonal phase relationships.

There are several additional issues worth considering regarding this interpolation process. If the amplitude loss, inherent in interpolation, of a factor of two is bothersome, we can make the final mixing 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 efficient polyphase filters for the lowpass filtering.

Finally, if it's sensible in your implementation, consider replacing the final adder with a multiplexer (because alternate samples of the sequences at Node D are zeros). In this case, the mixing sequence in the bottom path would be changed to 0,-1,0,1.

Used with the permission of the publisher, Prentice Hall, this on-going series of articles on Embedded.com is based on copyrighted material from "Understanding Digital Signal Processing, Second Edition" by Richard G. Lyons. The book can be purchased on line.

Richard Lyons is a consulting systems engineer and lecturer with Besser Associates. As a lecturer with Besser and an instructor for the University of California Santa Cruz Extension, Lyons has delivered digitasl signal processing seminars and training course at technical conferences as well at companies such as Motorola, Freescale, Lockheed Martin, Texas Instruments, Conexant, Northrop Grumman, Lucent, Nokia, Qualcomm, Honeywell, National Semiconductor, General Dynamics and Infinion.