1. Signals, I/Q, Sampling, and Spectra

Start With a Physical Wave

An antenna receives one rapidly changing voltage. The receiver shifts the wanted band to baseband and records it as two coordinates:

• I (in-phase): one axis of the complex signal.
• Q (quadrature): the axis 90 degrees from I.
• Physical meaning: together they retain amplitude and phase; they are not two independent radio waves.

$$x[n]=I[n]+jQ[n]$$

• $x[n]$: complex baseband sample at sample index $n$.
• $I[n]$, $Q[n]$: measured in-phase and quadrature coordinates.
• $j$: imaginary unit; it labels the perpendicular Q axis.

For a simple complex tone,

$$x[n]=A e^{j(2\pi f_b n/f_s+\phi)}$$

• $A$: tone amplitude; larger $A$ makes a larger I/Q circle.
• $f_b$: baseband tone frequency in hertz.
• $f_s$: sample rate in samples per second; $n$ is the sample index.
• $\phi$: starting phase in radians; $j$ is the imaginary unit.
• Read the plot: radius tracks amplitude; angle tracks phase. One plot alone cannot identify why either changed.

References for this section2

• MathWorks, OFDM Resource Grid and Channel Estimation.
• MathWorks, Communications Toolbox Test and Measurement.

Sampling: What Can Be Preserved?

• Sampling: recording a waveform at regular time intervals.
• Aliasing: different input frequencies appear identical after sampling too slowly.
• Anti-alias filter: removes unwanted out-of-band energy before analogue-to-digital conversion.
• Always state: occupied bandwidth, filtering, and whether samples are real or complex.

For a real low-pass signal with highest frequency $B$:

$$f_s>2B$$

• $f_s$: sample rate in samples per second.
• $B$: highest input frequency in hertz for this real low-pass case.
• Complex baseband: a suitably filtered band of width $B$ can use a rate around $B$ under the stated convention; filter transition margin and frequency placement still matter.

References for this section2

• MathWorks, OFDM Resource Grid and Channel Estimation.
• MathWorks, Communications Toolbox Test and Measurement.

Worked Example: A Tone in Three Views

Take a 1 MHz baseband tone sampled at 8 million complex samples per second:

1. I/Q rotation: phase advances $2\pi(1/8)=\pi/4$, or 45 degrees, per sample. Eight samples complete one turn.
2. FFT view: with $N=1024$ samples, the spacing between plotted frequency bins is

$$\Delta f=\frac{f_s}{N}=\frac{8\ \mathrm{MHz}}{1024}=7.8125\ \mathrm{kHz}$$

• $\Delta f$: spacing between FFT frequency bins.
• $f_s$: sample rate; here 8 million complex samples/s.
• $N$: number of measured samples in the transform; here 1024.
• Important: bin spacing is not automatically the ability to resolve two nearby tones. Window, observation time, and SNR also matter; zero-padding adds no measured time.

Idea	Physical meaning	Where it helps
Sample rate	How often the receiver records the waveform	Preserving the desired bandwidth and setting processing cost
I/Q amplitude and phase	Instantaneous complex position of the baseband signal	Modulation, channel estimation, and beamforming
Frequency spectrum	How signal energy is distributed across frequency	Detecting occupied band, interference, and filtering needs
Filter response	Frequencies passed, attenuated, or delayed by a system	Rejecting unwanted energy without damaging the wanted waveform

References for this section2

• MathWorks, OFDM Resource Grid and Channel Estimation.
• MathWorks, Communications Toolbox Test and Measurement.

FFT: Another View of the Same Samples

• DFT: turns one finite block of time samples into complex frequency bins.

• FFT: an efficient algorithm for computing that transform.

• Spectrum peak: may be a wanted tone or interference; read it with sample rate, window, duration, and normalization.

• Useful filtering: removes out-of-band noise or interference.

• Too narrow: cuts away part of the wanted waveform.

• Group delay: a filter can also distort timing; compare spectra before and after at the same reference level.

The DFT of a block $x[n]$ is

$$X[k]=\sum_{n=0}^{N-1}x[n]e^{-j2\pi kn/N}$$

• $X[k]$: complex value of frequency bin $k$; its magnitude and phase both matter.
• $x[n]$: input sample $n$; $N$: number of samples.
• $n$, $k$: time-sample and frequency-bin indices; $j$ is the imaginary unit.
• Leakage: a finite observation spreads a non-bin-centred tone across bins; a window can reduce sidelobes but widens the main lobe.

References for this section2

• MathWorks, OFDM Resource Grid and Channel Estimation.
• MathWorks, Communications Toolbox Test and Measurement.

Application: Debug an SDR Capture

If an SDR constellation looks rotated or spread:

1. Verify RF centre frequency, gain, sample rate, and clipping.
2. Inspect the spectrum for frequency offset or adjacent interference.
3. Inspect phase over time: steady rotation suggests carrier offset; irregular motion can indicate phase noise or a changing channel.
4. Only then judge the equalizer or decoder.

• Keep complex I/Q: power alone discards phase needed for delay estimation, beamforming, and localization.
• Record metadata: sample rate, calibration, time reference, and antenna position.

References for this section2

• MathWorks, OFDM Resource Grid and Channel Estimation.
• MathWorks, Communications Toolbox Test and Measurement.

What to Check in an Experiment

1. Confirm sample rate, centre frequency, gain setting, and whether samples are real or complex.
2. Inspect the time-domain envelope for clipping, dropouts, and burst timing.
3. Inspect the spectrum for occupied bandwidth, spurs, and interfering energy.
4. Check I/Q balance and constellation shape before interpreting a channel estimate.

• Next in the receiver: OFDM and receiver processing.
• Physical background: spectrum, frequency, and bandwidth.

References for this section2

• MathWorks, OFDM Resource Grid and Channel Estimation.
• MathWorks, Communications Toolbox Test and Measurement.