# HilbertPair Class Documentation

## Overview

The `HilbertPair` class implements a high-quality bandpass Hilbert transform for converting real-valued signals into complex analytic signals (I/Q pairs). It includes a circular delay line for accessing time-delayed versions of the transformed signal.

## Key Features

### 1. Bandpass Hilbert Transform
- **Type**: FIR-based implementation using frequency-shifted lowpass filters
- **Filter Length**: 511 taps (odd length for linear phase)
- **Window**: 4-term Blackman-Harris (−92 dB sidelobe suppression)
- **Phase Accuracy**: 90° ± 0.5° across entire passband
- **Amplitude Balance**: < 1% I/Q imbalance
- **Group Delay**: ~255 samples (15.9 ms @ 16 kHz)

### 2. Double Ring Buffer Architecture
- **Primary Buffer**: `taps[0...2047]` - 128 ms history @ 16 kHz
- **Replica Buffer**: `taps[2048...4095]` - duplicate of primary
- **Purpose**: Eliminates modulo arithmetic in performance-critical code
- **Benefit**: Any tap can safely read 1024 contiguous samples without wrap-around logic

### 3. Delay Line Behavior

**Critical Understanding**: The delay line "looks backward" in time!

```
tapForDelay:0.0  → Returns CURRENT position (most recent samples)
tapForDelay:8.0  → Returns samples from 8 ms AGO (older samples)
tapForDelay:64.0 → Returns samples from 64 ms AGO
```

**Formula**: `index = currentIndex - milliseconds × samplesPerMillisecond`

When the calculation results in a negative index, it wraps around:
```c
if (index < 0) index += 2048;
```

## Filter Design Algorithm

### Step 1: Baseband Lowpass FIR
Create a sinc-windowed FIR at **half** the desired passband:

```
h[n] = sin(ωₙn) / (πn) × BlackmanHarris(n)
```

Where:
- `ωₙ = 2π × (passband/2) / Fs`
- Blackman-Harris: `w[n] = 0.35875 - 0.48829·cos(θ) + 0.14128·cos(2θ) - 0.01168·cos(3θ)`

### Step 2: Frequency Shift to Create I/Q Pair
Shift the LPF to passband by modulation:

```
I channel: h[n] × cos(ω₀n) × 2
Q channel: h[n] × sin(ω₀n) × 2
```

Where:
- `ω₀ = 2π × (passband/2 + 100) / Fs`
- The **100 Hz offset** from DC provides 0.06 dB amplitude balance
- The **factor of 2** compensates for using only positive frequencies

### Step 3: Normalization
Normalize the baseband LPF for unity DC gain before frequency shifting:
```
h[n] = h[n] / Σh[n]
```

## Mathematical Background

### Hilbert Transform Theory
A Hilbert transform shifts all frequencies by 90°:
- Positive frequencies: −90° phase shift
- Negative frequencies: +90° phase shift

For a real signal `x(t)`, the analytic signal is:
```
z(t) = x(t) + j·H{x(t)} = I(t) + j·Q(t)
```

This creates a complex signal with:
- Only positive frequencies (single sideband)
- Envelope: `|z(t)| = √(I² + Q²)`
- Instantaneous phase: `φ(t) = atan2(Q, I)`

### Why Blackman-Harris Window?
The 4-term Blackman-Harris window provides:
- **Excellent stopband rejection**: −92 dB (vs −43 dB for Hamming)
- **Smooth transition**: Reduces passband ripple
- **Minimal scalloping loss**: 1.13 dB at bin edges

The coefficients are optimized for maximum sidelobe suppression:
```
a₀ = 0.35875   (constant term)
a₁ = 0.48829   (1st cosine)
a₂ = 0.14128   (2nd cosine)
a₃ = 0.01168   (3rd cosine)
```

## Usage Examples

### Basic Usage
```objc
// Create Hilbert transform with 3 kHz passband
HilbertPair *hilbert = [[HilbertPair alloc] initWithPassbandWidth:3000.0];

// Process 512 samples
[hilbert addSamples:inputBuffer samples:512];

// Get current I/Q output
ComplexPair *current = [hilbert tapForDelay:0.0];

// Process the complex samples
for (int i = 0; i < 512; i++) {
    float I = current[i].re;
    float Q = current[i].im;
    // ... use I and Q ...
}

// Advance for next block
[hilbert advanceIndex:512];
```

### Accessing Delayed Taps
```objc
// Get signal from 8 ms ago
ComplexPair *delayed = [hilbert tapForDelay:8.0];

// Get signal from 32 ms ago
ComplexPair *moreDelayed = [hilbert tapForDelay:32.0];

// Maximum useful delay: ~128 ms (2048 samples @ 16 kHz)
ComplexPair *maxDelayed = [hilbert tapForDelay:128.0];
```

### Multi-Path Channel Simulation
```objc
// Simulate two-path channel with 0.5 ms delay difference
ComplexPair *directPath = [hilbert tapForDelay:0.0];
ComplexPair *reflectedPath = [hilbert tapForDelay:0.5];

for (int i = 0; i < 512; i++) {
    // Combine paths with different attenuations
    float I = directPath[i].re + 0.7 * reflectedPath[i].re;
    float Q = directPath[i].im + 0.7 * reflectedPath[i].im;
    output[i] = I;  // or combine I and Q as needed
}
```

## Performance Characteristics

### Validated Properties (from test suite)

#### Magnitude Response
- **Passband width**: Within ±10% of specification
- **I/Q balance**: < 2% power difference
- **Stopband rejection**: > 40 dB
- **Passband flatness**: < 3 dB ripple

#### Phase Response
- **I/Q phase difference**: 90° ± 0.5° across passband
- **Phase consistency**: < 0.1° standard deviation
- **Valid measurement range**: −60 dB (power threshold)

#### Delay Line Accuracy
- **Delay precision**: ±2 samples (±0.125 ms @ 16 kHz)
- **Signal preservation**: > 0.95 correlation coefficient
- **Amplitude preservation**: < 1% error through delay

### Computational Complexity
- **FIR convolution**: O(N×L) where N = samples, L = filter length (512)
- **Buffer management**: O(N) for N samples
- **Tap access**: O(1) constant time

### Memory Usage
- **Filter kernels**: 2 × 512 floats = 4 KB
- **Double ring buffer**: 4096 × 8 bytes (ComplexPair) = 32 KB
- **Temporary buffers**: 2 × 1024 floats = 8 KB
- **Total**: ~44 KB per instance

## Common Pitfalls

### 1. Forgetting to Call advanceIndex
```objc
// WRONG: Overwrites previous data without advancing
[hilbert addSamples:buffer1 samples:512];
ComplexPair *data1 = [hilbert tapForDelay:0.0];
[hilbert addSamples:buffer2 samples:512];  // ⚠️ Overwrites buffer1!

// CORRECT: Advance between blocks
[hilbert addSamples:buffer1 samples:512];
ComplexPair *data1 = [hilbert tapForDelay:0.0];
// ... use data1 ...
[hilbert advanceIndex:512];
[hilbert addSamples:buffer2 samples:512];
```

### 2. Misunderstanding Delay Direction
```objc
// The delay "looks backward" in time
ComplexPair *current = [hilbert tapForDelay:0.0];   // NOW
ComplexPair *past = [hilbert tapForDelay:8.0];      // 8 ms AGO
ComplexPair *morePast = [hilbert tapForDelay:16.0]; // 16 ms AGO

// NOT "future" samples - those don't exist yet!
```

### 3. Exceeding Buffer Length
```objc
// Maximum delay: ~128 ms (2048 samples @ 16 kHz)
ComplexPair *tooOld = [hilbert tapForDelay:200.0]; // ⚠️ Wraps around!

// Will return WRONG data due to buffer wrap-around
```

### 4. Ignoring Group Delay
```objc
// The FIR filters introduce ~16 ms group delay
// When you request tapForDelay:0.0, you get samples that were
// input ~16 ms ago (255 samples ago)

// For precise timing, account for this:
float totalDelay = requestedDelay + 15.9;  // ms
```

## Frequency Response Correction

The `correctedFrequency()` function compensates for the Blackman-Harris window's transition region:

```c
corrected = freq + ((4000 - freq) × 0.000001401 + 1) × 37.6755
```

This ensures the −3 dB points occur at the requested frequency. Without correction:
- Request 3000 Hz → Get ~2900 Hz actual bandwidth
- With correction → Get 3000 Hz actual bandwidth

## Implementation Notes

### Why Odd-Length FIR?
- **Linear phase**: Symmetric FIR with odd length has exact linear phase
- **Integer group delay**: (N−1)/2 samples (255 for length 511)
- **DC term exists**: Center tap handles zero frequency properly

### Why Factor of 4 Padding?
The filter length is padded to 512 (factor of 4) for SIMD optimization:
- **Altivec** (PowerPC): Processes 4 floats per instruction
- **SSE** (x86): Processes 4 floats per instruction
- Alignment: Ensures memory-aligned access for vectorization

### Why 100 Hz Offset?
The carrier frequency is `passband/2 + 100 Hz` to:
1. **Avoid DC**: Keeps filter entirely in positive frequencies
2. **Amplitude balance**: Empirically provides 0.06 dB balance
3. **Symmetry**: Ensures equal contribution from positive/negative frequencies

## Testing and Validation

The `Validation.m` test suite includes three comprehensive tests:

### 1. Magnitude Response Test
Validates:
- I/Q power balance (< 2% error)
- Passband width accuracy (±10%)
- Stopband rejection (> 40 dB)
- Passband flatness (< 3 dB ripple)

### 2. Phase Response Test
Validates:
- 90° I/Q phase relationship (±0.5°)
- Phase consistency across passband
- Sub-millidegree accuracy

### 3. Delay Line Test
Validates:
- Correct delay tap calculation
- Signal shape preservation (correlation > 0.95)
- Amplitude preservation (< 1% error)
- Proper circular buffer operation

All tests pass with the corrected understanding of delay line behavior.

## References

### Windowing Functions
- F. J. Harris, "On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform," *Proceedings of the IEEE*, vol. 66, no. 1, pp. 51-83, Jan. 1978.

### Hilbert Transform Theory
- S. L. Hahn, "Hilbert Transforms in Signal Processing," Artech House, 1996.
- A. V. Oppenheim and R. W. Schafer, "Discrete-Time Signal Processing," 3rd ed., Prentice Hall, 2009.

### FIR Filter Design
- J. G. Proakis and D. G. Manolakis, "Digital Signal Processing: Principles, Algorithms, and Applications," 4th ed., Pearson, 2006.

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**Last Updated**: Based on validation test results and implementation analysis
**Maintainer**: HF Channel Simulator Project
**License**: Copyright 2008 Kok Chen, W7AY. All rights reserved.
