Signal filtering and spectra

Smoothing a noisy series and finding its rhythm

A filter turns one series into another by mixing each value with its neighbours. A moving average of $N$ days is the simplest filter: each output is the average of the last $N$ inputs, so noise shrinks but the output also lags behind. An exponential moving average (EMA) gives recent days more weight: $y_t=\alpha x_t+(1-\alpha)y_{t-1}$ , where $\alpha=2/(N+1)$ is the smoothing weight. signal.lfilter(b, a, x) runs any such filter on a series, using only past and present values. The average delay of an EMA is $(1-\alpha)/\alpha$ days, and that of an $N$ -day moving average is $(N-1)/2$ days, so a 19-day average and an EMA with $\alpha=0.1$ both lag by about 9 days.

The second job is finding cycles. signal.periodogram(x) splits the variance of a series into frequencies, measured in cycles per day. A peak at frequency $f$ means a repeating pattern with period $1/f$ days. With one observation per day you can see frequencies only up to $0.5$ cycles per day, the Nyquist limit, because a faster cycle cannot be told apart from a slower one. One warning matters a great deal in trading. signal.filtfilt runs the filter forward and then backward, which removes the lag, but each output then uses future values. A signal built with filtfilt looks perfect in a backtest and cannot be traded, because tomorrow's data is not available today. Use only causal filters such as lfilter for anything you plan to trade.

y_t=\alpha x_t+(1-\alpha)y_{t-1},\quad \alpha=\frac{2}{N+1},\quad \text{lag}=\frac{1-\alpha}{\alpha}\ \ \Big(\text{moving average: }\frac{N-1}{2}\Big),\qquad \text{period}=\frac1f

(1)

Three numbers and a cycleTake the series

10,12,11

with

\alpha=0.5

and start the EMA at the first value. Then

y_0=10

y_1=0.5\cdot12+0.5\cdot10=11

and

y_2=0.5\cdot11+0.5\cdot11=11

. A 3-day moving average of

10,12,11,13

gives

11

and then

12

. A 19-day window means

\alpha=2/20=0.1

and a lag of

(1-0.1)/0.1=9

days. A pure cycle that repeats

0.05

times per day has a period of

1/0.05=20

days, and the periodogram peaks exactly there. On real data, SPY's daily returns have a lag-1 autocorrelation of only

-0.10

, but their squares have

0.27

at lag 1 and

0.30

at lag 5: big days follow big days, which is volatility clustering. An EMA of squared returns with weight

\alpha=0.06

(RiskMetrics

\lambda=0.94

) reached

0.75\%

a day on 21 August 2026 (

11.9\%

a year), well below the full-sample

1.20\%

. That is exactly what a slow-moving average of the past can tell you without looking ahead.

Smoothing costs lag: a longer window or a smaller $\alpha$ removes more noise and delays the signal more.
Periods from peaks: a periodogram peak at frequency $f$ means a cycle of $1/f$ days, up to a limit of 0.5 cycles a day.
Only causal filters for trading: filtfilt uses the future, so it is for analysis, never for a live signal.

Lab · what's graded

An EMA step10 pts
A moving average10 pts
The weight for a window10 pts
A period from a frequency10 pts
The lag of an EMA10 pts
What each routine does15 pts
Why filtfilt cannot be traded35 pts

Smoothing a noisy series and finding its rhythm

y_t=\alpha x_t+(1-\alpha)y_{t-1},\quad \alpha=\frac{2}{N+1},\quad \text{lag}=\frac{1-\alpha}{\alpha}\ \ \Big(\text{moving average: }\frac{N-1}{2}\Big),\qquad \text{period}=\frac1f

(1)

Three numbers and a cycleTake the series

10,12,11

with

\alpha=0.5

and start the EMA at the first value. Then

y_0=10

y_1=0.5\cdot12+0.5\cdot10=11

and

y_2=0.5\cdot11+0.5\cdot11=11

. A 3-day moving average of

10,12,11,13

gives

11

and then

12

. A 19-day window means

\alpha=2/20=0.1

and a lag of

(1-0.1)/0.1=9

days. A pure cycle that repeats

0.05

times per day has a period of

1/0.05=20

days, and the periodogram peaks exactly there. On real data, SPY's daily returns have a lag-1 autocorrelation of only

-0.10

, but their squares have

0.27

at lag 1 and

0.30

at lag 5: big days follow big days, which is volatility clustering. An EMA of squared returns with weight

\alpha=0.06

(RiskMetrics

\lambda=0.94

) reached

0.75\%

a day on 21 August 2026 (

11.9\%

a year), well below the full-sample

1.20\%

. That is exactly what a slow-moving average of the past can tell you without looking ahead.

Smoothing costs lag: a longer window or a smaller $\alpha$ removes more noise and delays the signal more.
Periods from peaks: a periodogram peak at frequency $f$ means a cycle of $1/f$ days, up to a limit of 0.5 cycles a day.
Only causal filters for trading: filtfilt uses the future, so it is for analysis, never for a live signal.

Lab · what's graded

An EMA step10 pts
A moving average10 pts
The weight for a window10 pts
A period from a frequency10 pts
The lag of an EMA10 pts
What each routine does15 pts
Why filtfilt cannot be traded35 pts

Smoothing a noisy series and finding its rhythm

Lab · what's graded

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Signal filtering and spectra

Smoothing a noisy series and finding its rhythm

Lab · what's graded

Log in to run your code.