7.0 Filtering Of Time Series

ATM 552 Notes:Filtering of Time Series Chapter 7page 2057.0 Filtering of Time Series7.1 IntroductionIn this section we will consider the problem of filtering time or space series so thatcertain frequencies or wavenumbers are removed and some are retained. This is an oftused and oft-abused method of accentuating certain frequencies and removing others.The technique can be used to isolate frequencies that are of physical interest from thosethat are not. It can be used to remove high frequency noise or low frequency trends fromtime series and leave unaltered the frequencies of interest. These applications are calledlow-pass and high-pass filtering, respectively. A band-pass filter will remove both highfrequencies and low frequencies and leave only frequencies in a band in the middle.Band-pass filters tend to make even noise look periodic, or at least quasi-periodic. Wewill begin by noting a few important theorems that constitute the fundamental tools ofnon-recursive filtering.The Convolution Theorem:If two functions f1(t) and f2(t) have Fourier transforms F1(ω) and F2(ω) then the Fouriertransform of f1(t) f2(t) is12πAnd the Fourier transform of F1(λ )F2 (ω λ)dλ f1 (τ ) f 2 (t τ ) dτ is F1(ω) F2(ω). This latter result is the most useful in filtering, since it says that theFourier transform of the convolution of two functions in time is just the product of theFourier transforms of the individual functions.Parseval’s Theorem: 1f1( t ) f2 (t )dt 2π F2 (ω ) F1(ω )* dω for f1(t) f2(t) f(t).Copyright 2016 Dennis L. Hartmann2/20/164:43 PM205

ATM 552 Notes:Filtering of Time Series Chapter 7 1f (t ) dt 2π2page 206 F(ω )2dω Here we see that the variance of a time series integrated over all time is equal to thepower spectrum integrated over all frequency.7.2 FilteringSuppose we wish to modify oscillations of certain frequencies in a time series whilekeeping other frequencies the same. e.g. remove high frequency oscillations (low-passfilter), remove low frequencies (high-pass filter), or both (band-pass filter). The ozonelayer of Earth’s atmosphere is a low-pass filter for sunlight in the sense that it absorbs allenergy with wavelengths shorter than 300 nm before it reaches the surface. A coupledifferent approaches to filtering can be taken.7.2.1 Fourier MethodFourier analyzed time series to compute amplitudes at all frequencies. Modify theseamplitudes as desired, then reconstitute the series.f ( t ) Cω i cos (ω i t φi )(7.1)iNf filtered (t) Cω i R(ω ) cos(ω i t φi )(7.2)i 1Here the function R(w) is the response function of the desired filtering process andmeasures the ratio of the amplitude of the filtered to the unfiltered time series as afunction of frequency.R (ω ) Cω filteredCω originalThe problem with this method is that the reconstructed time series may not resemble theoriginal one, particularly near the ends. This is the same general characteristic offunctional fits discussed in an earlier chapter. Also, you need the whole record of databefore you can produce a single filtered data point, and the most recently acquired valuesare at the end of the data stream, where the problems with the Fourier method are worst.So if you want to do real-time filtering, Fourier methods are hopeless. For real-timeproblems the recursive methods are quite good.Copyright 2016 Dennis L. Hartmann2/20/164:43 PM206

ATM 552 Notes:Filtering of Time Series Chapter 7page 2077.2.2 Centered, Non-recursive Weighting MethodIn the centered non-recursive weighting method, the time series is subjected to aweighted running average, so that the filtered point is a weighted sum of surroundingpoints.ffiltered ( t ) J wk f (t kΔt)(7.3)k JSome data points will be lost from each end of the time series since we do not have thevalues to compute the smoothed series for i J and i N-J.It seems obvious that such an operation can most reasonably produce only smoothed timeseries and hence constitutes a low-pass filter. However, a high-pass filter can beconstructed quite simply by subtracting the low-pass filtered time series from the originaltime series. The new high-pass response function will then beRH (ω ) 1 RL (ω )(7.4)Where the subscripts H and L refer to high- and low-pass filters. One can then design ahigh-pass filter by first designing a low-pass filter that removes just those frequencies onewishes to retain. You can also make a band-pass filter by applying a low pass filter to atime series that has already been high-passed (or vice versa), in which case the responsefunction is the product of the two response functions (center case below). Or you cansubtract a low pass filtered version of the data set from another one with a cutoff at ahigher frequency, as illustrated below on the right.1RH 1-RLRB RH RLRB RL1 - RL211RHR(ω)R(ω)R(ω)RL0ωRB0RL2RL1RBω0ω7.3 The Response FunctionThe response function is the spectrum of amplitude modifications made to all frequenciesby the filtering function.Copyright 2016 Dennis L. Hartmann2/20/164:43 PM207

ATM 552 Notes:Filtering of Time Series Chapter 7R(ω ) page 208amplitudeof output series at frequency ωamplitude of input series at frequency ωR 2 (ω ) Φ(ω ) output power at ω Φ(ω ) input power at ωR(ω) can be real or imaginary. Some filtering is done by nature and instruments andthese may introduce phase errors. Any filter we would design and apply would have areal response function, unless we desire to introduce a phase shift. Phase shifting filtersare not too commonly used in meteorological or oceanographic data analysis ormodeling, and so we will not discuss them except in the context of recursive filtering,where a phase shift is often introduced with a single pass of a recursive filter.How do we design a weighting with the desired frequency response? Our smoothingoperation can be writteng(t ) J f (t kΔt) w( kΔt)(7.5)k Jwhere g(t) is the smoothed time series, f(t) is the original time series and w(k t) is theweighting. In the continuous case we can write this as g(t ) f (τ ) w( t τ )dτ(7.6) The filtered output g(t) is just the convolution of the unfiltered input series f(t) and thefilter weighting function w(t). From the convolution theorem the Fourier transform of f (τ ) w(t τ )dτisF(ω ) W (ω ) so thatG(ω ) F(ω ) W(ω )i.e. to obtain the frequency spectrum or Fourier coefficients of the output we multiply theFourier transform of the input times the Fourier transform of the weighting function.Copyright 2016 Dennis L. Hartmann2/20/164:43 PM208

ATM 552 Notes:Filtering of Time Series Chapter 7Pg (ω ) G(ω )G* (ω ) F (ω )W (ω ) ( F(ω )W (ω ))page 209* F (ω ) F* (ω ) W (ω )W * (ω ) F (ω ) 2 W (ω ) 2Simple Example:Suppose our input time series consists of a single cosine wave of amplitude 1.The output signal is theng( t ) K wk cos(ω (t kΔt ))k 1And the Fourier Transform G(ω ) 2 g(t ) cosω t dt0isG (ω ) 2 wk cos (ω t ω kΔt ) cos ω t dtκ wk cos ω kΔtk wk cos ω t kkNow F(ω) 1 so thatR (ω ) G (ω ) wk cos ω t k W (ω ) as k .F (ω ) k The Response function R(ω) is just the Fourier transform of the weighting function w(t).If we assume that the response function and the weighting function are symmetric,Copyright 2016 Dennis L. Hartmann2/20/164:43 PM209

ATM 552 Notes:Filtering of Time Series Chapter 7 R (ω ) W (ω ) 2 w ( t ) cos ωτ dτpage 210τ kΔt0and conversely the weighting function can be obtained from a specified (desired)Response function w (τ ) 2 R (ω ) cos ωτ dω0 w(τ) and R(ω) constitute a Fourier Transform Pair.Some examples:A Bad Example: The rectangular “Boxcar” weighting function or “running mean”smoother.1on the interval 0 τ Tw(τ ) TBoxcar Weighting Function1/T00TtAs we recall, the Fourier transform of the boxcar is the sinc function ωT sin 2 R (ω ) ωT2This response function approaches one as ωT/2 approaches zero. Hence it has no effecton frequencies with periods that are very long compared to the averaging interval T.When ωT 2n π, n 1,2,3., R(ω) 0. Τhe running mean smoother exactly removeswavelengths of which the length of the running mean is an exact multiple. For example,a 12 month running mean removes the annual cycle and all its higher harmonics.Copyright 2016 Dennis L. Hartmann2/20/164:43 PM210

ATM 552 Notes:Filtering of Time Series Chapter 7page 2111.0sinc Response0.50.-0.5Note that the response function, R(ω), of the running mean smoother is negative in theintervals 2π, -4π, 6π, -8π, etc. This means that frequencies in these intervals will bephase shifted by 180 because of the negative side lobes and the slow, 1/(ωT), drop off ofthe amplitude of these side lobes the “running mean” filter is not very good. It may beuseful if we are sure that the variance for ωT π is small.7.4 Inverse ProblemSuppose we wish to design a filter with a very sharp cutoffPerfect Square Response Function1.0R(ω)ωωc From w(τ ) 2 R(ω ) cos ωτ dω0Copyright 2016 Dennis L. Hartmann2/20/164:43 PM211

ATM 552 Notes:Filtering of Time Series Chapter 7page 212 ω τ sin c 2 w(τ ) ω cτ2we get,This is a damped sine wave (sinc function) again.The first zero crossing occurs atω cτ2π1 π τ 2ω c fcThe fact that the weight extends across several of these periods before dropping to zerocauses severe problems at the beginning and end of a finite time series. In practice it isdesirable to settle for a less sharp cutoff of R(ω) for which more practical weightingfunctions are available.“Practical” Filters(1) “Gaussian bell”e xThe function2forms a Fourier Transform pair with itself. Hence a Gaussian bell weighting functionproduces a Gaussian bell frequency response. Although, of course, when you considerthe response function the bell is peaked at zero frequency and we usually ignore thenegative frequencies.(2) The Triangular One-Two-One filtering functiong( t ) 111f (t τ ) f ( t ) f (t τ )424ωR(ω ) cos 2v Δt 2 where ν is the number of times the filter is applied.The response function for the 1-2-1 smoother approaches a Gaussian bell shape for v 5,R(ω) 0.Copyright 2016 Dennis L. Hartmann2/20/164:43 PM212

ATM 552 Notes:Filtering of Time Series Chapter 7page 213(3) Compromise Square Response:You can construct a filter which (1) has relatively few weights, (2) Has a “sort of” squareresponse and (3) has fairly small negative side lobes.Compromise Response Function1.0R(ω)BCAωThe larger the ratio A/B the larger side lobes one must accept. Some weights arenegative.You can find a number of these in the literature and in numerous software packages. Inthe next section we will show how to construct centered, non-recursive filter weights toorder.Copyright 2016 Dennis L. Hartmann2/20/164:43 PM213

ATM 552 Notes:Filtering of Time Series Chapter 7page 2147.5 Construction of Symmetric Nonrecursive Filters7.5.1 Fourier Construction of filter weightsIn this section we will describe methods for the construction of simple non-recursivefilters. Suppose we consider a simple symmetric non-recursive filterNyn Cxk n kwhereCk C k(7.7)k NTime-Shifting Theorem:To perform a Fourier transform of (7.7), it is useful to first consider the timeshifting theorem. Suppose we wish to calculate the Fourier transform of a time series f(t),which has been shifted by a time interval Δt a. Begin by substituting into the Fourierintegral representation of f(t).f (t a) 1 F(ω ) e iω (t a) dω2 π f (t a) 1 iωaeF(ω ) e iωt dω 2π(7.8)From (7.10) we infer that the Fourier transform of a time series shifted by a time intervalis equal to the Fourier transform of the unshifted time series multiplied by the factor,z eiωΔt(7.9)We can Fourier transform (7.7) and use the time shifting theorem (7.8) to obtain N iω kΔt Y (ω ) Ck eX (ω ) k N (7.10)Where Y(ω) and X(ω) are the Fourier transforms of y(t) and x(t). Because Ck C-k ande ix e ixcos x 2(7.11)we can write (7.10) asR (ω ) Copyright 2016 Dennis L. HartmannNY (ω ) C0 2 Ck cos (ω kΔt )X (ω )k 12/20/164:43 PM(7.12)214