iir

iir(X, b, a)

Applies an infinite impulse response (IIR) filter to the input data X and returns the filtered result. An IIR filter uses previous output values as feedback, allowing them to achieve sharper transition bands with a much lower filter order.

The mathematical expression is: a[0]·y[n] = (b[0]·x[n] + b[1]·x[n-1] + … + b[M]·x[n-M]) - (a[1]·y[n-1] + … + a[N]·y[n-N]), where b denotes the numerator coefficients and a denotes the denominator coefficients.

X is a regular vector of numeric type specifying the data to be filtered.

b is a non-empty regular vector of numeric type specifying the numerator coefficients of the IIR filter.

a is a non-empty regular vector of numeric type specifying the denominator coefficients of the IIR filter.

Returns a result of the same form and length as the input X.

Example 1. Apply an IIR filter to a vector.

x = 1 2 3 4 5
b = 0.2 0.2
a = 1 -0.5
iir(x, b, a)
// output: [0.2,0.7,1.35,2.075,2.8375]

x = [1, 2, NULL, 4]  // equivalent to [1, 2, 0, 4]
b = 0.2 0.2
a = 1 -0.5
iir(x, b, a)
// output: [0.2,0.7,0.75,1.175]

Example 2. The moving average crossover strategy is one of the most classic trend-following strategies in quantitative trading. It uses a fast EMA and a slow EMA: a buy signal is generated when the fast EMA crosses above the slow EMA, and a sell signal is generated when the fast EMA crosses below the slow EMA. An EMA is essentially a first-order IIR filter.

Compared with an FIR filter that provides a similar degree of smoothing, an IIR filter requires only two coefficients. This offers significant computational advantages when backtesting strategies across hundreds of stocks in a broad market universe.

The following example simulates 1-minute bar data for a single trading day of a stock. Two first-order IIR filters, alphaFast (corresponding to a 10-period EMA) and alphaSlow (corresponding to a 30-period EMA), are used to extract the fast and slow trend lines, respectively, and generate crossover signals.

// 1. Create the catalog, database, and table
if(!existsCatalog("FinanceData")){
    createCatalog("FinanceData")
}
go
use catalog FinanceData

if(!existsDatabase("FinanceData.kline")){
    create database kline
    partitioned by VALUE(2026.06.01..2026.06.30),
    engine='TSDB'
}
go
if(!existsTable("FinanceData.kline", "min1")){
    create table kline.min1 (
        SecurityID SYMBOL,
        TradeDate DATE,
        BarTime TIMESTAMP,
        Open DOUBLE,
        High DOUBLE,
        Low DOUBLE,
        Close DOUBLE,
        Volume LONG
    )
    partitioned by TradeDate,
    sortColumns = [`SecurityID, `BarTime],
    keepDuplicates = ALL
}

// 2. Generate sample data: simulate 240 one-minute bars for 600519.SH on a trading day
n = 240
idx = 0..(n - 1)

baseTime = 2026.06.02T09:30:00.000
barTime = baseTime + idx * 60000

trendComponent = 1800.0 + 10.0 * sin(2.0 * pi * idx / n * 2.0) + cumsum(rand(0.2, n) - 0.1)
noise = norm(0.0, 1.0, n)

closePrice = round(trendComponent + noise, 2)
openPrice  = round(closePrice + norm(0.0, 0.5, n), 2)

highBase = iif(openPrice > closePrice, openPrice, closePrice)
lowBase  = iif(openPrice < closePrice, openPrice, closePrice)

highPrice = round(highBase + abs(norm(0.0, 0.3, n)), 2)
lowPrice  = round(lowBase  - abs(norm(0.0, 0.3, n)), 2)

volume = long(rand(10000, n) + 100)

t = table(
    symbol(take("600519.SH", n)) as SecurityID,
    take(2026.06.02, n) as TradeDate,
    barTime as BarTime,
    openPrice as Open,
    highPrice as High,
    lowPrice as Low,
    closePrice as Close,
    volume as Volume
)

// 3. Insert the data
kline.min1.append!(t)

// 4. Load the bar data
raw = select BarTime, Close
      from kline.min1
      where SecurityID = `600519.SH
        and TradeDate = 2026.06.02
      order by BarTime

// 5. Compute the fast and slow EMA lines using IIR filters
alphaFast = 2.0 / (10 + 1)
alphaSlow = 2.0 / (30 + 1)

emaResult = select
                BarTime,
                Close,
                iir(Close, [alphaFast], [1.0, -(1.0 - alphaFast)]) as emaFast,
                iir(Close, [alphaSlow], [1.0, -(1.0 - alphaSlow)]) as emaSlow
            from raw

// 6. Generate crossover signals
signals = select
              BarTime,
              Close,
              emaFast,
              emaSlow,
              iif(
                  emaFast > emaSlow and prev(emaFast) <= prev(emaSlow),
                  "BUY",
                  iif(
                      emaFast < emaSlow and prev(emaFast) >= prev(emaSlow),
                      "SELL",
                      "HOLD"
                  )
              ) as signal
          from emaResult
          order by BarTime

// 7. Query all buy and sell signals
select *
from signals
where signal != "HOLD"
order by BarTime

Related Functions: fir