Using filtfilt helps for this problem:
using DSP, Plots
t = 1:1000; fs = 1/5; f0 = 1/750 # sampling and cutoff frequencies
rawdata = 500*(exp.(-t/500) .+ 0.1*rand(length(t))) # noisy input data
data_filt = filtfilt(digitalfilter(Lowpass(f0, fs=fs), Butterworth(4)), rawdata)
plot(rawdata,label="input")
plot!(data_filt, lc=:blue, lw=2,label="filtered using filtfilt")
