IQ Samples

Each IQ sample is a single complex number where the real part is the In-Phase (I) component and the imaginary part is the Quadrature (Q) component. This mathematical formulation preserves both amplitude and phase information simultaneously.

• Instantaneous amplitude: √(I² + Q²)
• Instantaneous phase: arctan(Q/I)
• Enables direct manipulation in the complex plane without loss of information
• Forms the native data type for software-defined radio (SDR) processing chains

The I and Q components are derived by mixing the incoming RF signal with two local oscillator signals that are exactly 90 degrees out of phase. This orthogonality ensures the two channels carry independent information.

• I channel: mixed with cos(2πf_c t)
• Q channel: mixed with sin(2πf_c t)
• Perfect orthogonality prevents cross-channel interference
• Any mismatch in the 90-degree phase shift introduces IQ imbalance, a hardware impairment that degrades classification accuracy

IQ samples must be captured at a rate that satisfies the Nyquist-Shannon sampling theorem relative to the signal's complex baseband bandwidth. For a bandpass signal of bandwidth B, the complex sampling rate must be at least B samples per second.

• Complex sampling provides twice the spectral efficiency of real sampling
• A 20 MHz LTE channel requires a minimum complex sample rate of 20 MSPS
• Undersampling introduces aliasing artifacts that distort the constellation structure
• Practical systems often oversample at 2-4x the symbol rate to preserve pulse shape features

A stream of IQ samples forms a discrete-time series where the ordering encodes the temporal evolution of the modulated waveform. This sequential nature makes IQ data inherently suitable for recurrent and transformer-based neural architectures.

• Each sample represents one point in the complex plane at a specific time instant
• Symbol-rate sampling yields one sample per transmitted symbol
• Oversampled streams capture inter-symbol transition trajectories
• The temporal correlation between adjacent samples is a key feature exploited by LSTM and self-attention mechanisms for modulation recognition

Real-world IQ samples contain non-ideal artifacts introduced by the receiver's analog front-end that must be accounted for in robust classification models. These impairments become part of the learned feature distribution.

• DC offset: a constant bias in the I or Q channel from local oscillator leakage
• IQ gain imbalance: amplitude mismatch between the I and Q paths
• Phase noise: random fluctuations in the local oscillator's phase
• Carrier frequency offset (CFO): residual frequency error after downconversion
• Training on synthetic data alone without impairment modeling leads to brittle classifiers that fail on real hardware captures

When fed into neural networks, IQ samples are typically structured as a 2 × N real-valued matrix or a 1 × N complex-valued vector, where N is the number of time samples in the observation window.

• Common input shape for CNNs: [batch_size, 2, sequence_length] with I and Q as separate channels
• For complex-valued networks: [batch_size, sequence_length] with native complex arithmetic
• Typical observation windows range from 128 to 1024 samples for modulation classification
• Longer sequences capture more temporal context but increase computational latency
• The choice of input length represents a trade-off between classification confidence and real-time processing constraints

The mathematical framework that underpins IQ samples. A bandpass signal is frequency-shifted to center at zero hertz, producing a complex-valued signal s(t) = I(t) + jQ(t). This representation preserves all amplitude and phase information of the original modulated waveform while enabling processing at dramatically lower sample rates. The in-phase (I) component is the real part, and the quadrature (Q) component is the imaginary part, forming a complex envelope that fully describes the modulation state.

A two-dimensional scatter plot of IQ samples on the complex plane, where the I component is the x-axis and the Q component is the y-axis. Each point represents a transmitted symbol at a specific sampling instant. The geometric arrangement of these clusters directly reveals the modulation scheme:

• QPSK: 4 distinct clusters at 90° intervals
• 16-QAM: 16 clusters in a 4x4 grid
• 8-PSK: 8 clusters on a circle Constellation diagrams are the primary visual input for CNN-based modulation classifiers.

The Nyquist-Shannon sampling theorem dictates that a bandlimited signal must be sampled at a rate at least twice its bandwidth to enable perfect reconstruction. For IQ samples, the complex sampling rate must exceed the signal's information bandwidth. Undersampling causes aliasing, where high-frequency components fold into the observed spectrum, corrupting the constellation diagram and making modulation classification unreliable. Practical systems often sample at 2.5x to 4x the symbol rate to preserve pulse shape characteristics.

Before transmission, digital symbols are filtered by a pulse-shaping filter (e.g., Root-Raised Cosine) to limit bandwidth. At the receiver, a matched filter maximizes SNR. The IQ samples captured between these filters contain the shaped pulses. Inter-Symbol Interference (ISI) occurs when pulse tails from adjacent symbols overlap at the sampling instant, distorting the constellation. Proper timing synchronization is critical to sample at the eye diagram's maximum opening to minimize ISI and enable accurate classification.

Real-world direct-conversion receivers introduce IQ imbalance: gain mismatch between I and Q branches and phase error deviating from the ideal 90° offset. This manifests as a skewed and rotated constellation that does not match the ideal modulation template. The received IQ samples become r(t) = α * s(t) + β * s*(t), where the conjugate term s*(t) creates an image interference. Deep learning classifiers trained on hardware-in-the-loop data can learn to be robust to these impairments without explicit compensation.

A critical distinction in IQ sample processing. The sample rate (fs) is the rate at which the analog-to-digital converter produces IQ pairs. The symbol rate (Rs) is the rate at which modulation symbols are transmitted. The ratio fs/Rs defines samples per symbol (SPS). A classifier's input tensor typically requires a fixed number of samples, so the SPS determines how many symbols are captured in one inference window. Higher SPS preserves more pulse shape detail but increases computational load. Typical values range from 2 to 8 SPS.