Перейти к содержанию

Лабораторная 3.3 — Digital mixing and frequency shift

Lab 3.3 — Digital Mixing and Frequency Shift

Goal

Shift a complex IQ signal in frequency using a numerically controlled oscillator (NCO) and complex multiplication.

This lab is the direct bridge between SDR software processing and FPGA blocks such as DDS/NCO, complex multiplier, frequency translator and digital downconverter.

Theory

Digital mixing multiplies the input complex signal by a complex exponential:

y[n] = x[n] * exp(j * 2*pi*f_shift*n/Fs)

For a complex baseband signal, this shifts the spectrum by f_shift.

Important concepts:

  • complex sinusoid;
  • NCO / DDS;
  • phase accumulator;
  • frequency resolution;
  • complex multiplication;
  • positive and negative frequency shifts;
  • spectral images and wraparound;
  • fixed-point phase and sine/cosine representation.

Experiment

Generate or load an IQ signal with a tone or modulated waveform located away from DC.

Then:

  1. estimate or define the input frequency offset;
  2. generate an NCO;
  3. multiply the signal by the NCO;
  4. compare spectra before and after frequency shift;
  5. verify that the target signal moves to the expected frequency;
  6. discuss NCO frequency resolution and fixed-point phase accumulator width.

Python implementation

Minimum expected script structure:

import numpy as np
import matplotlib.pyplot as plt

fs = 2.4e6
n = 32768
t = np.arange(n) / fs

f0 = 420e3
x = np.exp(1j * 2*np.pi*f0*t)

f_shift = -420e3
nco = np.exp(1j * 2*np.pi*f_shift*t)
y = x * nco

freq = np.fft.fftshift(np.fft.fftfreq(n, d=1/fs))
X = np.fft.fftshift(np.fft.fft(x * np.hanning(n)))
Y = np.fft.fftshift(np.fft.fft(y * np.hanning(n)))

plt.figure()
plt.plot(freq/1e3, 20*np.log10(np.maximum(np.abs(X), 1e-12)), label="before")
plt.plot(freq/1e3, 20*np.log10(np.maximum(np.abs(Y), 1e-12)), label="after")
plt.grid(True)
plt.xlabel("Frequency, kHz")
plt.ylabel("Magnitude, dB")
plt.legend()
plt.show()

MATLAB implementation

Minimum expected script structure:

fs = 2.4e6;
N = 32768;
t = (0:N-1).' / fs;

f0 = 420e3;
x = exp(1j*2*pi*f0*t);

fShift = -420e3;
nco = exp(1j*2*pi*fShift*t);
y = x .* nco;

freq = fftshift((-floor(N/2):ceil(N/2)-1).' * fs / N);
X = fftshift(fft(x .* hann(N)));
Y = fftshift(fft(y .* hann(N)));

figure; hold on;
plot(freq/1e3, 20*log10(max(abs(X), 1e-12)), 'DisplayName', 'before');
plot(freq/1e3, 20*log10(max(abs(Y), 1e-12)), 'DisplayName', 'after');
grid on;
xlabel('Frequency, kHz');
ylabel('Magnitude, dB');
legend('Location', 'best');

C++ bridge

A future C++ primitive should separate NCO generation from complex multiplication:

struct NcoState {
    uint64_t phase;
    uint64_t phase_step;
};

std::complex<float> next_nco_sample(NcoState& nco);
std::complex<float> mix_sample(std::complex<float> x, std::complex<float> osc);

Validation tests:

  • generated frequency matches target shift;
  • phase continuity is preserved;
  • positive and negative shifts are correct;
  • mixed tone lands at expected FFT bin;
  • fixed-point NCO error is bounded.

FPGA / Verilog bridge

Hardware mapping:

Function RTL block
phase accumulation NCO phase accumulator
sine/cosine generation LUT, CORDIC or vendor DDS
complex multiply four multipliers or optimized three-multiplier form
output scaling fixed-point rounding and saturation

Streaming interface target:

clk, rst
in_valid,  in_i,  in_q
out_valid, out_i, out_q

NCO frequency resolution

For an accumulator of width A bits:

df = Fs / 2^A
Accumulator width Frequency resolution
24 bit Fs / 16,777,216
32 bit Fs / 4,294,967,296
48 bit Fs / 281,474,976,710,656

Expected plots

Produce at least:

  1. spectrum before mixing;
  2. spectrum after mixing;
  3. zoomed view around target frequency;
  4. optional phase accumulator trace;
  5. optional NCO quantization error plot.

Report checklist

  • [ ] State Fs, input tone frequency and target shift.
  • [ ] Show the complex NCO equation.
  • [ ] Plot spectrum before and after mixing.
  • [ ] Verify the final frequency location.
  • [ ] Explain positive vs negative frequency shifts.
  • [ ] Calculate NCO resolution for at least 32-bit phase accumulator.
  • [ ] Explain how the mixer maps to FPGA.
  • [ ] Discuss fixed-point scaling and saturation.

Engineering conclusion template

Digital mixing shifts the signal from ____ kHz to ____ kHz. In FPGA, the same
operation becomes an NCO plus complex multiplier. The main hardware trade-offs
are phase accumulator width, sine/cosine generation method, multiplier count and
output scaling.