First, some assumptions about the road and our car:

```
road_len_m <- 1000 # road length
speed_kmh <- 80 # car speed in km/h
sample_rate_hz <- 200 # sampling rate of the 3D accelerometer
speed_ms <- speed_kmh / 3.6 # car speed in m/s
sample_len <- round(speed_ms / sample_rate_hz, digits = 2) # sample size
num_samples <- round(road_len_m / sample_len) # how many samples we collected
GRAVITY_ACCEL <- 9.80665 # ms^-2
print(sample_len)
```

```
## [1] 0.11
```

```
print(num_samples)
```

```
## [1] 9091
```

First, we use some sample data obrained from a 3D-accelerometer. We need to trim NA valued from the signal, because there might be gaps (NA values) and the interpolation doesn't work with NAs.

```
# random signal with simulated gravity
random_accz <- (rnorm(num_samples) - GRAVITY_ACCEL)
# random gaps in the signal to later demonstrate gaps interpolation
random_gaps <- sapply(rnorm(num_samples),
function(x){ifelse(abs(x) > .7, 1, NA)})
signal <- data.frame(
sampleid = seq_len(num_samples),
dist_meters = seq(from = 0, to = road_len_m, by = sample_len),
accZ = random_accz * random_gaps
)
head(signal$accZ, num_samples) -> signal$accZ_orig
plot(signal$dist_meters, signal$accZ, type = "o", pch = "+", cex = .5,
main = "Accelerometer signal (original)",
xlab = "Distance traveled [m]",
ylab = expression( paste("Z-acceleration [", m * s ^ -2, "]") ))
```

```
# signal without the standard gravity acceleration
(signal$accZ_orig - GRAVITY_ACCEL) -> signal$accZ_nogravity
# z-score normalization (subtracting mean and dividing by sd)
scale(signal$accZ_orig) -> signal$accZ
# removing leading and trailing NAs in the whole matrix
na.trim(signal) -> signal
plot(signal$dist_meters, signal$accZ, type = "o", pch = "+", cex = .5,
main = "Accelerometer signal (z-score normalized)",
xlab = "Distance traveled [m]",
ylab = expression( paste("Z-acceleration [", m * s ^ -2, "]") ))
```

Now, we look at the gaps closely:

```
head(signal, 500) -> signal_head
plot(signal_head$dist_meters, signal_head$accZ,
type = "o", xlab = NA, ylab = NA, pch = "+", cex = .5,
main = paste("First", nrow(signal_head), "samples with gaps"))
```

We need to interpolate the values between the gaps:

```
na.approx(signal$accZ, na.rm = FALSE) -> signal$accZ_approx
head(signal, 500) -> signal_head
plot(signal_head$dist_meters, signal_head$accZ,
type = "p", pch = "+", cex = .5, xlab = NA, ylab = NA,
main = paste("First", nrow(signal_head), "samples interpolated"))
lines(signal_head$dist_meters, signal_head$accZ_approx,
col = "red", pch = ".", xlab = NA, ylab = NA)
```

We can also analyze frequency content of the signal by using **Continuous
Wavelet Transform (CWT)**. The following plot is called “scaleogram”.

```
w <- wt(cbind(signal$sampleid, signal$accZ_approx), dj = 1/2)
plot(w)
```

We can extract the CWT coeficients representing certaing frequency bands.
The `power.corr`

matrix represents bias-correction version.

```
nscales <- nrow(w$power.corr)
signal$cwt_mid <- w$power.corr[floor(.5 * nscales),]
signal$cwt_high <- w$power.corr[floor(.2 * nscales),]
signal$cwt_low <- w$power.corr[floor(.8 * nscales),]
plot(signal$cwt_high, type = "l")
lines(signal$cwt_mid, col = "blue", lw = 4)
lines(signal$cwt_low, col = "red", lw = 4)
```

Here, we compute moving average and root mean squared value:

```
rollmean(signal$accZ_approx, k = 10, fill = NA) -> signal$rollmean10
rollmean(signal$accZ_approx, k = 20, fill = NA) -> signal$rollmean20
rms <- function(x) sqrt(mean(x^2)) # same as `rms` from `seewave` package
rollapply(signal$accZ_approx, width = 20, fill = NA, FUN = rms ) -> signal$rms20
```

```
head(signal, 3000) -> signal_head
plot(signal_head$dist_meters,
signal_head$accZ_approx,
type = "l", xlab = NA, ylab = NA,
main = paste("First", nrow(signal_head), "samples interpolated"))
lines(signal_head$dist_meters, signal_head$rollmean10, col = "red", lw = 3)
lines(signal_head$dist_meters, signal_head$rollmean20, col = "blue", lw = 3)
lines(signal_head$dist_meters, signal_head$rms20, col = "green", lw = 3)
```

h