QuanTAlib/lib/cycles/dsp/dsp.md

DSP: Detrended Synthetic Price

"Remove the trend, reveal the cycles."

The Detrended Synthetic Price (DSP) indicator, developed by John Ehlers, is a cycle analysis tool that removes trend components to expose underlying price cycles. By differencing two exponential moving averages (fast and slow), DSP creates a zero-centered oscillator that highlights momentum shifts.

Historical Context

John Ehlers introduced the Detrended Synthetic Price as part of his cycle analysis toolkit. The indicator builds on the MACD concept but uses EMA periods derived from cycle theory: quarter-cycle (fast) and half-cycle (slow) lengths. This mathematical relationship helps isolate cycle components while suppressing trend noise.

The "synthetic" in the name refers to how DSP synthesizes a detrended view of price by subtracting the slower-reacting EMA from the faster one. When the fast EMA exceeds the slow EMA, price momentum is bullish; when below, momentum is bearish.

Unlike traditional oscillators that bound between fixed levels, DSP oscillates around zero with amplitude proportional to price volatility and cycle strength.

Architecture & Physics

DSP uses dual EMA smoothing with bias correction during warmup to produce accurate values from the first bar.

Core Components

1. Period Parameter: Base cycle length (default 40)
2. Fast EMA: Smoothing with period = max(2, round(period/4)) - quarter cycle
3. Slow EMA: Smoothing with period = max(3, round(period/2)) - half cycle
4. Bias Correction: Warmup decay factors eliminate EMA initialization bias
5. State Record: Maintains EMA values and warmup factors for rollback support

Period Derivation

For period = 40:

• Fast period = max(2, round(40/4)) = 10
• Slow period = max(3, round(40/2)) = 20

For period = 4 (minimum):

• Fast period = max(2, round(4/4)) = max(2, 1) = 2
• Slow period = max(3, round(4/2)) = max(3, 2) = 3

Calculation Flow

For each update:

1. Calculate fast alpha: \alpha_f = 2 / (p_f + 1)
2. Calculate slow alpha: \alpha_s = 2 / (p_s + 1)
3. Update fast EMA with bias correction
4. Update slow EMA with bias correction
5. DSP = corrected_fast_ema - corrected_slow_ema

Mathematical Foundation

EMA Alpha Calculation

\alpha = \frac{2}{period + 1}

For fast period 10: \alpha_f = \frac{2}{11} \approx 0.1818

For slow period 20: \alpha_s = \frac{2}{21} \approx 0.0952

EMA Update (with bias correction)

The raw EMA recursion:

EMA^{raw}_t = \alpha \cdot P_t + (1 - \alpha) \cdot EMA^{raw}_{t-1}

The warmup decay factor tracks bias:

e_t = (1 - \alpha) \cdot e_{t-1}

Starting with e_0 = 1, this converges to 0 as the EMA warms up.

The bias-corrected EMA:

EMA_t = \frac{EMA^{raw}_t}{1 - e_t}

DSP Formula

DSP_t = EMA^{fast}_t - EMA^{slow}_t

where both EMAs are bias-corrected.

Properties

• Range: Unbounded, oscillates around zero
• Zero Crossing: Indicates momentum shift
• Positive Values: Fast EMA > Slow EMA (bullish momentum)
• Negative Values: Fast EMA < Slow EMA (bearish momentum)
• Warmup: IsHot when e_{slow} < 0.05 (5% remaining bias)

Example Calculation

For period = 40 with constant price 100:

After warmup, both EMAs converge to 100:

• Fast EMA = 100
• Slow EMA = 100
• DSP = 100 - 100 = 0

For uptrend (price rising steadily):

• Fast EMA responds quicker, stays closer to current price
• Slow EMA lags behind
• DSP > 0 (positive momentum)

Performance Profile

Metric	Score	Notes
Throughput	~8 ns/bar	O(1) constant time
Allocations	0	Zero-allocation in hot path
Complexity	O(1)	Fixed operations per update
Accuracy	10	Exact EMA with bias correction

Operation Count (per update)

Operation	Count	Notes
ADD/SUB	~6	EMA updates and DSP calculation
MUL	~6	Alpha multiplications
DIV	2	Bias correction divisions
FMA	4	Fused multiply-add for EMA

Quality Metrics

Metric	Score	Notes
Accuracy	10/10	Exact EMA with bias correction
Timeliness	8/10	Faster than traditional MACD
Overshoot	7/10	EMA smoothing reduces overshoot
Smoothness	8/10	Dual EMA provides good smoothing

Validation

Library	Status	Notes
TA-Lib	N/A	Not available in TA-Lib
Skender	N/A	Not available in Skender
Tulip	N/A	Not available in Tulip
PineScript	✅	Validated against original DSP implementation

DSP is validated through mathematical properties:

• Constant price produces zero DSP
• Uptrend produces positive DSP
• Downtrend produces negative DSP
• Oscillates around zero for cyclic price patterns

Common Pitfalls

1. Period Selection: The period parameter represents the dominant cycle length. Use half the detected cycle period for optimal results. Default 40 works for daily data.

2. Comparison to MACD: DSP differs from MACD in period derivation. MACD uses arbitrary 12/26 periods; DSP uses cycle-theory-based period/4 and period/2.

3. Warmup Behavior: DSP includes bias correction, so early values are usable. IsHot indicates when the slow EMA bias drops below 5%.

4. Amplitude Interpretation: DSP amplitude scales with price level. A $1 stock and $100 stock with identical percentage moves will have 100x different DSP amplitudes.

5. Zero Crossings: Not all zero crossings are tradeable. Use in conjunction with cycle analysis or additional confirmation.

6. Trending Markets: In strong trends, DSP stays positive or negative for extended periods. Cycle analysis is most effective in ranging markets.

Usage

using QuanTAlib;

// Create a 40-period DSP indicator
var dsp = new Dsp(period: 40);

// Update with new values
var result = dsp.Update(new TValue(DateTime.UtcNow, 100.0));

// Access the last calculated DSP value
Console.WriteLine($"DSP: {dsp.Last.Value}");

// Chained usage
var source = new TSeries();
var dspChained = new Dsp(source, period: 40);

// Static batch calculation
var output = Dsp.Calculate(source, period: 40);

// Span-based calculation
Span<double> outputSpan = stackalloc double[source.Count];
Dsp.Batch(source.Values, outputSpan, period: 40);

Applications

Cycle Detection

DSP zero crossings help identify cycle turning points:

• DSP crosses above zero: cycle trough (potential buy)
• DSP crosses below zero: cycle peak (potential sell)

Trend Filtering

Use DSP sign to filter trades with trend direction:

• DSP > 0: Only take long trades
• DSP < 0: Only take short trades

Momentum Confirmation

DSP slope confirms momentum strength:

• Rising DSP: Increasing bullish momentum
• Falling DSP: Increasing bearish momentum

Divergence Analysis

Like other oscillators, DSP divergences signal potential reversals:

• Price higher high, DSP lower high: bearish divergence
• Price lower low, DSP higher low: bullish divergence

Comparison to Related Indicators

DSP vs MACD

Feature	DSP	MACD
Period basis	Cycle theory (P/4, P/2)	Arbitrary (12, 26)
Signal line	None (optional)	9-period EMA
Bias correction	Yes	No
Histogram	No	Yes (MACD - Signal)

DSP vs Detrended Price Oscillator (DPO)

Feature	DSP	DPO
Calculation	Fast EMA - Slow EMA	Price - SMA shifted
Time alignment	Current	Shifted back period/2 + 1
Leading/Lagging	Leading	Centered (neither)

References

• Ehlers, J.F. (2001). Rocket Science for Traders. Wiley.
• Ehlers, J.F. (2004). Cybernetic Analysis for Stocks and Futures. Wiley.
• TradingView PineScript: DSP implementation in cycle analysis scripts.