From f60f22d5230bb935b6906e68feea4be07ca27b61 Mon Sep 17 00:00:00 2001
From: brunozc <brunozuada@gmail.com>
Date: Fri, 4 Jul 2025 10:01:22 +0200
Subject: [PATCH 1/3] fix: improve time padding logic in TimeSignalProcessing
 and add tests for time interpolation

---
 SignalProcessingTools/time_signal.py | 2 +-
 tests/test_time_signal.py            | 6 ++++++
 2 files changed, 7 insertions(+), 1 deletion(-)

diff --git a/SignalProcessingTools/time_signal.py b/SignalProcessingTools/time_signal.py
index fdd961c..d9283bd 100644
--- a/SignalProcessingTools/time_signal.py
+++ b/SignalProcessingTools/time_signal.py
@@ -110,7 +110,7 @@ def __init__(self,
             # pad signal at the end if necessary to get full windows
             if signal_length % window_size != 0:
                 self.signal = np.append(self.signal, np.zeros(window_size - (signal_length % window_size)))
-                self.time = np.append(self.time, np.zeros(window_size - (signal_length % window_size)))
+                self.time = np.append(self.time, self.time[-1]+np.cumsum(np.ones(window_size - (signal_length % window_size))*(1/Fs)))
                 self.operations.append(f"Signal padded with zeros (original length: {signal_length}, new length: {len(self.signal)})")
 
     def __str__(self) -> str:
diff --git a/tests/test_time_signal.py b/tests/test_time_signal.py
index 15b9cac..b9bbe90 100644
--- a/tests/test_time_signal.py
+++ b/tests/test_time_signal.py
@@ -4,6 +4,7 @@
 from SignalProcessingTools.time_signal import TimeSignalProcessing, IntegrationRules, Windows
 
 TOL = 3e-3
+FLOAT_TOL = 1e-12
 FREQ = 6
 AMP = 1.75
 
@@ -215,6 +216,11 @@ def test_psd(test_data):
     np.testing.assert_almost_equal(sig.frequency_Pxx[np.argmax(sig.Pxx)], FREQ, 2)
     assert (np.abs((np.max(sig.Pxx) - peak_psd) / peak_psd) < 0.035)
 
+    # test the time interpolation
+    assert len(sig.time) == np.ceil(50001/4000)*4000
+    assert ((np.diff(sig.time)-1/500) < FLOAT_TOL).all()
+
+
 def test_v_eff():
     """
     Test the v_eff function

From d527ee41fb341f9f478b4bb7430b1c30e43c9ea2 Mon Sep 17 00:00:00 2001
From: brunozc <brunozuada@gmail.com>
Date: Fri, 4 Jul 2025 10:02:24 +0200
Subject: [PATCH 2/3] feat: add pre-commit configuration and update
 requirements for pre-commit package

---
 .pre-commit-config.yaml | 10 ++++++++++
 requirements.txt        |  3 ++-
 setup.cfg               |  3 ++-
 3 files changed, 14 insertions(+), 2 deletions(-)
 create mode 100644 .pre-commit-config.yaml

diff --git a/.pre-commit-config.yaml b/.pre-commit-config.yaml
new file mode 100644
index 0000000..bfd963e
--- /dev/null
+++ b/.pre-commit-config.yaml
@@ -0,0 +1,10 @@
+repos:
+  - repo: https://github.com/google/yapf
+    rev: v0.43.0
+    hooks:
+      - id: yapf
+        name: yapf
+        language: python
+        entry: yapf
+        args: [-i]
+        types: [python]
diff --git a/requirements.txt b/requirements.txt
index 906b331..7e435cc 100644
--- a/requirements.txt
+++ b/requirements.txt
@@ -2,4 +2,5 @@ numpy==2.2.5
 scipy==1.15.0
 matplotlib==3.10.0
 pytest==8.0.2
-tox==4.13.0
\ No newline at end of file
+tox==4.13.0
+pre_commit==4.2.0
\ No newline at end of file
diff --git a/setup.cfg b/setup.cfg
index f6ab469..47b6a59 100644
--- a/setup.cfg
+++ b/setup.cfg
@@ -23,4 +23,5 @@ python_requires = >=3.10
 [options.extras_require]
 testing =
     pytest>=8.0
-    tox>=4.13
\ No newline at end of file
+    tox>=4.13
+    pre_commit>=4.2.0
\ No newline at end of file

From 771e469b9892bb43b0b399169349809397e8b15e Mon Sep 17 00:00:00 2001
From: brunozc <brunozuada@gmail.com>
Date: Fri, 4 Jul 2025 10:02:47 +0200
Subject: [PATCH 3/3] applied linter

---
 SignalProcessingTools/space_signal.py |  97 ++++++++----
 SignalProcessingTools/time_signal.py  | 204 ++++++++++++++++++--------
 example_space_signal.py               |   5 +-
 example_time_signal.py                |  38 +++--
 tests/test_space_signal.py            |  35 +++--
 tests/test_time_signal.py             | 144 ++++++++++++------
 6 files changed, 349 insertions(+), 174 deletions(-)

diff --git a/SignalProcessingTools/space_signal.py b/SignalProcessingTools/space_signal.py
index 0d96cd2..138124e 100644
--- a/SignalProcessingTools/space_signal.py
+++ b/SignalProcessingTools/space_signal.py
@@ -8,7 +8,11 @@ class SpaceSignalProcessing:
     """
     SignalProcessing class for processing signals in space.
     """
-    def __init__(self, x: npt.NDArray[np.float64], values: npt.NDArray[np.float64], Fs: Optional[float] = None):
+
+    def __init__(self,
+                 x: npt.NDArray[np.float64],
+                 values: npt.NDArray[np.float64],
+                 Fs: Optional[float] = None):
         """
         Initializes the ProcessSignal object.
 
@@ -40,7 +44,6 @@ def __init__(self, x: npt.NDArray[np.float64], values: npt.NDArray[np.float64],
         self.max_fast = None
         self.max_fast_Dx = None
 
-
     def compute_track_longitudinal_levels(self):
         """
         Computes the track longitudinal levels, following EN 13848-1:2006.
@@ -53,20 +56,34 @@ def compute_track_longitudinal_levels(self):
         - D3: 70m < lambda <= 150m (1/150 Hz < f <= 1/70 Hz)
         """
 
-        sig = TimeSignalProcessing(self.coordinates, self.signal_raw, Fs=self.fs)
-        sig.filter([1/5., 1.], 4, type_filter="bandpass", filter_design=FilterDesign.BUTTERWORTH)
+        sig = TimeSignalProcessing(self.coordinates,
+                                   self.signal_raw,
+                                   Fs=self.fs)
+        sig.filter([1 / 5., 1.],
+                   4,
+                   type_filter="bandpass",
+                   filter_design=FilterDesign.BUTTERWORTH)
         self.d0 = sig.signal
 
         sig.reset()
-        sig.filter([1/25., 1/3.], 4, type_filter="bandpass", filter_design=FilterDesign.BUTTERWORTH)
+        sig.filter([1 / 25., 1 / 3.],
+                   4,
+                   type_filter="bandpass",
+                   filter_design=FilterDesign.BUTTERWORTH)
         self.d1 = sig.signal
         sig.reset()
 
-        sig.filter([1/70., 1/25.], 4, type_filter="bandpass", filter_design=FilterDesign.BUTTERWORTH)
+        sig.filter([1 / 70., 1 / 25.],
+                   4,
+                   type_filter="bandpass",
+                   filter_design=FilterDesign.BUTTERWORTH)
         self.d2 = sig.signal
         sig.reset()
 
-        sig.filter([1/150., 1/70.], 4, type_filter="bandpass", filter_design=FilterDesign.BUTTERWORTH)
+        sig.filter([1 / 150., 1 / 70.],
+                   4,
+                   type_filter="bandpass",
+                   filter_design=FilterDesign.BUTTERWORTH)
         self.d3 = sig.signal
         sig.reset()
 
@@ -82,22 +99,23 @@ def compute_Hmax(self, convert_m2mm: bool = True):
         """
 
         # octave bands used for the processing
-        one_third_octave_bands = [[.08, .10],
-                                  [.10, .126],
-                                  [.126, .16],
-                                  [.16, .20],
-                                  [.20, .253],
-                                  [.253, .32],
-                                  [.32, .40],
-                                  [.40, .50],
-                                  [.50, .63],
-                                  ]
-
+        one_third_octave_bands = [
+            [.08, .10],
+            [.10, .126],
+            [.126, .16],
+            [.16, .20],
+            [.20, .253],
+            [.253, .32],
+            [.32, .40],
+            [.40, .50],
+            [.50, .63],
+        ]
 
         # setting for the processing
         self.DXmaxFast = 1
         nb_fft_min = 256  # minimum number of samples for the power spectral density
-        derivative = [0, 0, 0, 2, 2, 2, 2, 2, 2]  # number of times that each frequency band is derived
+        derivative = [0, 0, 0, 2, 2, 2, 2, 2,
+                      2]  # number of times that each frequency band is derived
 
         # RMS of the square root of the power spectral density
         self.rms_bands = np.zeros(len(one_third_octave_bands))
@@ -111,26 +129,34 @@ def compute_Hmax(self, convert_m2mm: bool = True):
             self.signal = self.signal_raw * 1000
 
         # compute the power spectral density
-        n_fft = int(np.max([2 ** (np.ceil(np.log2(len(self.signal)))), nb_fft_min]))
+        n_fft = int(
+            np.max([2**(np.ceil(np.log2(len(self.signal)))), nb_fft_min]))
         # if signal is odd length, add a zero to make it even
         if len(self.signal) % 2 != 0:
             signal = np.append(self.signal, 0)
-            coordinates = np.append(self.coordinates, self.coordinates[-1] + (self.coordinates[1] - self.coordinates[0]))
+            coordinates = np.append(
+                self.coordinates, self.coordinates[-1] +
+                (self.coordinates[1] - self.coordinates[0]))
         else:
             signal = self.signal
             coordinates = self.coordinates
-        sig = TimeSignalProcessing(coordinates, signal, Fs=self.fs, window=Windows.HAMMING,
+        sig = TimeSignalProcessing(coordinates,
+                                   signal,
+                                   Fs=self.fs,
+                                   window=Windows.HAMMING,
                                    window_size=len(signal))
         sig.psd(nb_points=n_fft, detrend=False)
 
         # compute the rsm psd
-        self.__rms_effective(sig.frequency_Pxx, sig.Pxx, one_third_octave_bands, derivative)
+        self.__rms_effective(sig.frequency_Pxx, sig.Pxx,
+                             one_third_octave_bands, derivative)
         # compute the effective values
         self.__effective_values(one_third_octave_bands, derivative)
 
-
-    def __rms_effective(self, frequency: npt.NDArray[np.float64], Pxx: npt.NDArray[np.float64],
-                        one_third_octave_bands: List[Tuple[float, float]], derivative: List[int]):
+    def __rms_effective(self, frequency: npt.NDArray[np.float64],
+                        Pxx: npt.NDArray[np.float64],
+                        one_third_octave_bands: List[Tuple[float, float]],
+                        derivative: List[int]):
         """
         Computes RMS square root of power spectral density
 
@@ -148,10 +174,13 @@ def __rms_effective(self, frequency: npt.NDArray[np.float64], Pxx: npt.NDArray[n
         for i, band in enumerate(one_third_octave_bands):
             # find indexes where the bands exist
             idx = np.where((frequency >= band[0]) & (frequency < band[1]))[0]
-            Pxx[idx] = (2 * np.pi * frequency[idx]) ** (2 * derivative[i]) * Pxx[idx]
+            Pxx[idx] = (2 * np.pi * frequency[idx])**(2 *
+                                                      derivative[i]) * Pxx[idx]
             self.rms_bands[i] = np.sqrt(np.sum(Pxx[idx] * delta_f))
 
-    def __effective_values(self, one_third_octave_bands: List[Tuple[float, float]], derivative: List[int]):
+    def __effective_values(self, one_third_octave_bands: List[Tuple[float,
+                                                                    float]],
+                           derivative: List[int]):
         """
         Computes the effective values of the signal
 
@@ -170,8 +199,13 @@ def __effective_values(self, one_third_octave_bands: List[Tuple[float, float]],
 
         for i, band in enumerate(one_third_octave_bands):
             derivative_value = derivative[i]
-            sig = TimeSignalProcessing(self.coordinates, self.signal, Fs=self.fs)
-            sig.filter(np.array(band), N=3, type_filter="bandpass", filter_design=FilterDesign.BUTTERWORTH)
+            sig = TimeSignalProcessing(self.coordinates,
+                                       self.signal,
+                                       Fs=self.fs)
+            sig.filter(np.array(band),
+                       N=3,
+                       type_filter="bandpass",
+                       filter_design=FilterDesign.BUTTERWORTH)
             new_signal = sig.signal
 
             while derivative_value != 0:
@@ -181,7 +215,8 @@ def __effective_values(self, one_third_octave_bands: List[Tuple[float, float]],
             ksi = np.linspace(0, n * tau, int(n * tau / dx + 1))
             g = fout * np.exp(-ksi / tau)
 
-            convoluted_signal = np.sqrt(np.convolve(new_signal**2, g) * dx / tau)
+            convoluted_signal = np.sqrt(
+                np.convolve(new_signal**2, g) * dx / tau)
             self.max_fast[i] = np.max(convoluted_signal)
             idx = np.floor((len(self.signal) - np.floor(self.DXmaxFast / dx)) / 2) + \
                   np.linspace(0, np.floor(self.DXmaxFast / dx)-1, int(np.floor(self.DXmaxFast / dx)))
diff --git a/SignalProcessingTools/time_signal.py b/SignalProcessingTools/time_signal.py
index d9283bd..52bedf3 100644
--- a/SignalProcessingTools/time_signal.py
+++ b/SignalProcessingTools/time_signal.py
@@ -14,6 +14,7 @@ class FilterDesign(Enum):
     CHEBYSHEV = 2
     ELLIPTIC = 3
 
+
 class IntegrationRules(Enum):
     """
     Integration rules
@@ -21,6 +22,7 @@ class IntegrationRules(Enum):
     TRAPEZOID = 1
     SIMPSON = 2
 
+
 class Windows(Enum):
     """
     Windows types
@@ -34,10 +36,12 @@ class Windows(Enum):
     RECTANGULAR = 'boxcar'
     TRIANG = 'triang'
 
+
 class TimeSignalProcessing:
     """
     Signal processing class for time signals
     """
+
     def __init__(self,
                  time: npt.NDArray[np.float64],
                  signal: npt.NDArray[np.float64],
@@ -71,8 +75,7 @@ def __init__(self,
         self.Sxx = None
         self.frequency_Sxx = None
         self.time_Sxx = None
-        self.fft_settings = {"nb_points": None,
-                             "half_representation": False}
+        self.fft_settings = {"nb_points": None, "half_representation": False}
         # Track operations performed on the signal
         self.operations = []
 
@@ -93,25 +96,38 @@ def __init__(self,
             self.use_window = False
         else:
             if window_size == 0:
-                raise ValueError("When using a window the `window_size` must be specified")
+                raise ValueError(
+                    "When using a window the `window_size` must be specified")
             if window_size % 2 != 0:
                 raise ValueError("Window length must be even")
             if window not in Windows:
-                raise ValueError(f"Window type {window} not supported. Available types: {list(Windows)}")
+                raise ValueError(
+                    f"Window type {window} not supported. Available types: {list(Windows)}"
+                )
             if window_size > signal_length:
-                raise ValueError(f"Window length ({window_size}) cannot be greater than signal length ({signal_length}).")
+                raise ValueError(
+                    f"Window length ({window_size}) cannot be greater than signal length ({signal_length})."
+                )
 
             self.window = self.__create_window(window, window_size)
             self.window_size = window_size
             self.window_type = window
-            self.nb_windows = int(np.ceil((signal_length / window_size) * 2 - 2))
+            self.nb_windows = int(
+                np.ceil((signal_length / window_size) * 2 - 2))
             self.use_window = True
 
             # pad signal at the end if necessary to get full windows
             if signal_length % window_size != 0:
-                self.signal = np.append(self.signal, np.zeros(window_size - (signal_length % window_size)))
-                self.time = np.append(self.time, self.time[-1]+np.cumsum(np.ones(window_size - (signal_length % window_size))*(1/Fs)))
-                self.operations.append(f"Signal padded with zeros (original length: {signal_length}, new length: {len(self.signal)})")
+                self.signal = np.append(
+                    self.signal,
+                    np.zeros(window_size - (signal_length % window_size)))
+                self.time = np.append(
+                    self.time, self.time[-1] + np.cumsum(
+                        np.ones(window_size - (signal_length % window_size)) *
+                        (1 / Fs)))
+                self.operations.append(
+                    f"Signal padded with zeros (original length: {signal_length}, new length: {len(self.signal)})"
+                )
 
     def __str__(self) -> str:
         """
@@ -148,7 +164,8 @@ def __str__(self) -> str:
         return "\n".join(info)
 
     @staticmethod
-    def __create_window(window_type: Windows, size: int) -> npt.NDArray[np.float64]:
+    def __create_window(window_type: Windows,
+                        size: int) -> npt.NDArray[np.float64]:
         """
         Create a window array of specified type and size
 
@@ -234,10 +251,10 @@ def fft(self,
             # peak amplitude of stationary sinusoids.
             spectrum_w[:, w] = np.fft.fft(signal_w, nfft) / normalise_fct
 
-
         self.amplitude = np.mean(np.abs(spectrum_w), axis=1)
         # self.phase = np.unwrap(np.angle(np.mean(spectrum_w, axis=1)))
-        self.phase = np.angle(np.mean(np.exp(1j * np.angle(spectrum_w)), axis=1))
+        self.phase = np.angle(
+            np.mean(np.exp(1j * np.angle(spectrum_w)), axis=1))
 
         # compute frequency
         self.frequency = np.linspace(0, 1, nfft) * self.Fs
@@ -249,9 +266,11 @@ def fft(self,
             self.phase = self.phase[:int(nfft / 2)]
 
         # FFT settings: needed to perform inverse FFT
-        self.fft_settings = {"nb_points": nfft,
-                             "half_representation": half_representation,
-                             "odd_length": odd_length}
+        self.fft_settings = {
+            "nb_points": nfft,
+            "half_representation": half_representation,
+            "odd_length": odd_length
+        }
 
         # Add to operations list
         op_info = f"FFT (points: {nfft}, half representation: {half_representation})"
@@ -275,7 +294,8 @@ def inv_fft(self):
             "Please compute FFT with full representation.")
 
         if self.use_window:
-            raise ValueError("Cannot perform inverse FFT on the windowed signal.")
+            raise ValueError(
+                "Cannot perform inverse FFT on the windowed signal.")
 
         # get FFT settings
         odd_length = self.fft_settings["odd_length"]
@@ -290,7 +310,8 @@ def inv_fft(self):
         # inverse of the FFT signal
         self.signal_inv = np.real(spectrum_inv) * len(spectrum)
         # time from frequency
-        self.time_inv = np.cumsum(np.ones(len(spectrum)) * 1 / self.Fs) - 1 / self.Fs
+        self.time_inv = np.cumsum(
+            np.ones(len(spectrum)) * 1 / self.Fs) - 1 / self.Fs
 
         if odd_length:
             # remove last sample
@@ -298,11 +319,17 @@ def inv_fft(self):
             self.time_inv = self.time_inv[:-1]
 
         # Add to operations list
-        self.operations.append("Inverse FFT" + (" with windowing" if self.use_window else ""))
-
-    def integrate(self, rule: IntegrationRules = IntegrationRules.TRAPEZOID,
-                  baseline: bool = False, moving: bool = False, hp: bool = False, ini_cond: float = 0.,
-                  fpass: float = 0.5, n: int = 6):
+        self.operations.append("Inverse FFT" +
+                               (" with windowing" if self.use_window else ""))
+
+    def integrate(self,
+                  rule: IntegrationRules = IntegrationRules.TRAPEZOID,
+                  baseline: bool = False,
+                  moving: bool = False,
+                  hp: bool = False,
+                  ini_cond: float = 0.,
+                  fpass: float = 0.5,
+                  n: int = 6):
         """
         Numerical integration of signal
 
@@ -322,9 +349,13 @@ def integrate(self, rule: IntegrationRules = IntegrationRules.TRAPEZOID,
 
         # integration rule
         if rule == IntegrationRules.TRAPEZOID:
-            self.signal = integrate.cumulative_trapezoid(self.signal, self.time, initial=ini_cond)
+            self.signal = integrate.cumulative_trapezoid(self.signal,
+                                                         self.time,
+                                                         initial=ini_cond)
         elif rule == IntegrationRules.SIMPSON:
-            self.signal = integrate.cumulative_simpson(self.signal, x=self.time, initial=ini_cond)
+            self.signal = integrate.cumulative_simpson(self.signal,
+                                                       x=self.time,
+                                                       initial=ini_cond)
         else:
             sys.exit("Integration rule not supported")
 
@@ -346,13 +377,18 @@ def integrate(self, rule: IntegrationRules = IntegrationRules.TRAPEZOID,
         if moving:
             op_details.append("moving average correction")
         if hp:
-            op_details.append(f"highpass filter (cutoff: {fpass} Hz, order: {n})")
+            op_details.append(
+                f"highpass filter (cutoff: {fpass} Hz, order: {n})")
 
         self.operations.append(f"Integration ({', '.join(op_details)})")
 
-
-    def filter(self, Fpass: float, N: int, filter_design: FilterDesign = FilterDesign.ELLIPTIC,
-               type_filter: str = "lowpass", rp: float = 0.01, rs: int = 60):
+    def filter(self,
+               Fpass: float,
+               N: int,
+               filter_design: FilterDesign = FilterDesign.ELLIPTIC,
+               type_filter: str = "lowpass",
+               rp: float = 0.01,
+               rs: int = 60):
         """
         Filter signal
 
@@ -377,11 +413,23 @@ def filter(self, Fpass: float, N: int, filter_design: FilterDesign = FilterDesig
 
         # design filter
         if filter_design == FilterDesign.ELLIPTIC:
-            z, p, k = signal.ellip(N, rp, rs, np.array(Fpass) / (self.Fs / 2), btype=type_filter, output='zpk')
+            z, p, k = signal.ellip(N,
+                                   rp,
+                                   rs,
+                                   np.array(Fpass) / (self.Fs / 2),
+                                   btype=type_filter,
+                                   output='zpk')
         elif filter_design == FilterDesign.BUTTERWORTH:
-            z, p, k = signal.butter(N, np.array(Fpass) / (self.Fs / 2), btype=type_filter, output='zpk')
+            z, p, k = signal.butter(N,
+                                    np.array(Fpass) / (self.Fs / 2),
+                                    btype=type_filter,
+                                    output='zpk')
         elif filter_design == FilterDesign.CHEBYSHEV:
-            z, p, k = signal.cheby1(N, rp, np.array(Fpass) / (self.Fs / 2), btype=type_filter, output='zpk')
+            z, p, k = signal.cheby1(N,
+                                    rp,
+                                    np.array(Fpass) / (self.Fs / 2),
+                                    btype=type_filter,
+                                    output='zpk')
 
         sos = signal.zpk2sos(z, p, k)
 
@@ -389,8 +437,8 @@ def filter(self, Fpass: float, N: int, filter_design: FilterDesign = FilterDesig
         self.signal = signal.sosfiltfilt(sos, self.signal)
 
         # Add to operations list
-        self.operations.append(f"Filter ({type_filter}, cutoff: {Fpass} Hz, order: {N})")
-
+        self.operations.append(
+            f"Filter ({type_filter}, cutoff: {Fpass} Hz, order: {N})")
 
     def psd(self, detrend: str = "linear", nb_points: Optional[int] = None):
         """
@@ -403,11 +451,15 @@ def psd(self, detrend: str = "linear", nb_points: Optional[int] = None):
         """
 
         if detrend not in ["linear", False]:
-            raise ValueError("Detrend method not supported. Available methods: ['linear', False]")
+            raise ValueError(
+                "Detrend method not supported. Available methods: ['linear', False]"
+            )
 
         # check if window is initialized
         if not self.use_window:
-            raise ValueError("No window defined. Please define a window when initialising SignalProcessing.")
+            raise ValueError(
+                "No window defined. Please define a window when initialising SignalProcessing."
+            )
 
         # if nb_points is None: nb_points is window length
         if nb_points is None:
@@ -416,12 +468,18 @@ def psd(self, detrend: str = "linear", nb_points: Optional[int] = None):
             nfft = nb_points
 
         # compute PSD using Welch method
-        self.frequency_Pxx, self.Pxx = signal.welch(self.signal, fs=self.Fs, nperseg=self.window_size, nfft=nfft,
-                                                    window=self.window_type.value, scaling='density', detrend=detrend)
+        self.frequency_Pxx, self.Pxx = signal.welch(
+            self.signal,
+            fs=self.Fs,
+            nperseg=self.window_size,
+            nfft=nfft,
+            window=self.window_type.value,
+            scaling='density',
+            detrend=detrend)
 
         # Add to operations list
-        self.operations.append(f"PSD (window: {self.window_type.name}, size: {self.window_size})")
-
+        self.operations.append(
+            f"PSD (window: {self.window_type.name}, size: {self.window_size})")
 
     def v_eff_SBR(self, n: int = 4, tau: float = 0.125):
         """
@@ -438,13 +496,12 @@ def v_eff_SBR(self, n: int = 4, tau: float = 0.125):
         qsi = np.linspace(0, n * tau, int(n * tau * self.Fs + 1))
         g = fout * np.exp(-qsi / tau)
 
-
         # Frequency weighting parameters
         v0 = 1 / 1000  # Reference velocity [m/s]
-        f0 = 5.6       # Reference frequency [Hz]
+        f0 = 5.6  # Reference frequency [Hz]
 
         # Handle even/odd signal length for FFT
-        if self.signal.shape[0]  % 2 != 0:
+        if self.signal.shape[0] % 2 != 0:
             nv1 = int(self.signal.shape[0] / 2 + 0.5)
             nv2 = int(self.signal.shape[0] / 2 - 0.5)
         else:
@@ -456,25 +513,25 @@ def v_eff_SBR(self, n: int = 4, tau: float = 0.125):
         freq = np.arange(df, (nv1 + 1) * df, df)
 
         # Create high-pass weighting filter (human perception curve)
-        Hv = (1 / v0) * 1 / (np.sqrt(1 + (f0 / freq) ** 2))
+        Hv = (1 / v0) * 1 / (np.sqrt(1 + (f0 / freq)**2))
         Hv = np.append(0, Hv)  # Add DC component
 
         # Create low-pass filter with 50 Hz cutoff
         cut_off_number = int(np.ceil(50 / df))
         if cut_off_number < nv1:
             Hv2 = np.zeros(Hv.shape[0])
-            Hv2[:cut_off_number+1] = 1
+            Hv2[:cut_off_number + 1] = 1
         else:
             Hv2 = np.ones(Hv.shape[0])
 
         # Applies the frequency weighting functions
         Fv = np.fft.fft(self.signal)
-        Fhv = Hv2 * Hv * Fv[:nv1+1]
+        Fhv = Hv2 * Hv * Fv[:nv1 + 1]
         Fv = np.append(Fhv, np.flipud(np.conj(Fhv[1:nv2])))
         v_eff = np.real(np.fft.ifft(Fv))
 
         # moving root-mean-square through convolution with the exponential decay function `g`
-        v_eff = np.sqrt( np.convolve(v_eff**2, g) * (1 / self.Fs) /tau)
+        v_eff = np.sqrt(np.convolve(v_eff**2, g) * (1 / self.Fs) / tau)
 
         self.v_eff = v_eff[:self.signal.shape[0]]
 
@@ -514,14 +571,19 @@ def spectrogram(self):
         Compute spectrogram of signal
         """
         # compute spectrogram
-        f, t, Sxx = signal.spectrogram(self.signal, fs=self.Fs, window=self.window_type.value,
-                                       nperseg=self.window_size, noverlap=self.window_size // 8)
+        f, t, Sxx = signal.spectrogram(self.signal,
+                                       fs=self.Fs,
+                                       window=self.window_type.value,
+                                       nperseg=self.window_size,
+                                       noverlap=self.window_size // 8)
         self.Sxx = Sxx
         self.frequency_Sxx = f
         self.time_Sxx = t
 
         # Add to operations list
-        self.operations.append(f"Spectrogram (nperseg: {self.window_size}, noverlap: {self.window_size // 8})")
+        self.operations.append(
+            f"Spectrogram (nperseg: {self.window_size}, noverlap: {self.window_size // 8})"
+        )
 
     def one_third_octave_bands(self):
         """
@@ -534,24 +596,31 @@ def one_third_octave_bands(self):
         # https://en.wikipedia.org/wiki/Octave_band
         initial_frequency_band_number = -20
         final_frequency_band_number = 33
-        names = ("10", "12.5", "16", "20", "25", "31.5", "40", "50", "63", "80", "100", "125", "160", "200", "250", "315", "400",
-                 "500", "630", "800", "1000", "1250", "1600", "2000", "2500", "3.150", "4000", "5000", "6300", "8000",
-                 "10000", "12500", "16000", "20000")
+        names = ("10", "12.5", "16", "20", "25", "31.5", "40", "50", "63",
+                 "80", "100", "125", "160", "200", "250", "315", "400", "500",
+                 "630", "800", "1000", "1250", "1600", "2000", "2500", "3.150",
+                 "4000", "5000", "6300", "8000", "10000", "12500", "16000",
+                 "20000")
 
         # compute centre frequencies of the bands
-        f_centre = 1000 * (2 ** (np.arange(initial_frequency_band_number, final_frequency_band_number) / 3))
-        f_upper = f_centre * (2 ** (1 / 6))
-        f_lower = f_centre / (2 ** (1 / 6))
+        f_centre = 1000 * (2**(np.arange(initial_frequency_band_number,
+                                         final_frequency_band_number) / 3))
+        f_upper = f_centre * (2**(1 / 6))
+        f_lower = f_centre / (2**(1 / 6))
 
         # sum the signal for the bands
         if (self.Pxx is None) and (self.amplitude is None):
-            raise ValueError("No PSD nor FFT computed. Please compute either first.")
+            raise ValueError(
+                "No PSD nor FFT computed. Please compute either first.")
 
         if self.Pxx is not None:
             # determine the frequency bands
-            idx = np.where((f_lower < np.max(self.frequency_Pxx)) & (f_upper > np.min(self.frequency_Pxx)))[0]
+            idx = np.where((f_lower < np.max(self.frequency_Pxx))
+                           & (f_upper > np.min(self.frequency_Pxx)))[0]
             if len(idx) == 0:
-                raise ValueError("No frequency bands found in the PSD. Please check the frequency bands.")
+                raise ValueError(
+                    "No frequency bands found in the PSD. Please check the frequency bands."
+                )
 
             delta_f = self.frequency_Pxx[1] - self.frequency_Pxx[0]
 
@@ -561,14 +630,19 @@ def one_third_octave_bands(self):
 
             for i, val in enumerate(idx):
                 self.octave_bands_Pxx[i] = float(names[val])
-                mask = (self.frequency_Pxx >= f_lower[val]) & (self.frequency_Pxx < f_upper[val])
-                self.octave_bands_Pxx_power[i] = np.sum(self.Pxx[mask] * delta_f)
+                mask = (self.frequency_Pxx
+                        >= f_lower[val]) & (self.frequency_Pxx < f_upper[val])
+                self.octave_bands_Pxx_power[i] = np.sum(self.Pxx[mask] *
+                                                        delta_f)
 
         if self.amplitude is not None:
             # determine the frequency bands
-            idx = np.where((f_lower < np.max(self.frequency)) & (f_upper > np.min(self.frequency)))[0]
+            idx = np.where((f_lower < np.max(self.frequency))
+                           & (f_upper > np.min(self.frequency)))[0]
             if len(idx) == 0:
-                raise ValueError("No frequency bands found in the FFT. Please check the frequency bands.")
+                raise ValueError(
+                    "No frequency bands found in the FFT. Please check the frequency bands."
+                )
 
             # compute the FFT for the bands
             self.octave_bands_fft = np.zeros(len(idx))
@@ -576,5 +650,7 @@ def one_third_octave_bands(self):
 
             for i, val in enumerate(idx):
                 self.octave_bands_fft[i] = float(names[val])
-                mask = (self.frequency >= f_lower[val]) & (self.frequency < f_upper[val])
-                self.octave_bands_fft_power[i] = np.sum(self.amplitude[mask]**2)
+                mask = (self.frequency >= f_lower[val]) & (self.frequency
+                                                           < f_upper[val])
+                self.octave_bands_fft_power[i] = np.sum(
+                    self.amplitude[mask]**2)
diff --git a/example_space_signal.py b/example_space_signal.py
index 2aae46b..7bb469e 100644
--- a/example_space_signal.py
+++ b/example_space_signal.py
@@ -3,14 +3,12 @@
 from SignalProcessingTools.space_signal import SpaceSignalProcessing
 import pickle
 
-
 # Create test data
 x = np.linspace(0, 100, 50001)
 omega = 2 * np.pi * 6
 y = 1.75 * np.sin(omega * x)
 y_noise = y + 0.01 * np.sin(120 * x)
 
-
 # Create a SpaceSignalProcessing object and demonstrate basic functionality
 print("----------------------------------------------")
 print("EXAMPLE 1: Track longitudinal level processing")
@@ -63,7 +61,6 @@
 plt.tight_layout()
 plt.show()
 
-
 # Example 2: Computing Hmax parameters
 print("------------------------------------")
 print("EXAMPLE 2: Computing Hmax parameters")
@@ -75,7 +72,7 @@
     0.002 * np.sin(2 * np.pi * 0.1 * x_track) +  # Long wavelength component
     0.001 * np.sin(2 * np.pi * 0.2 * x_track) +  # Medium wavelength component
     0.0005 * np.sin(2 * np.pi * 0.4 * x_track) +  # Short wavelength component
-    0.0002 * np.random.randn(len(x_track))        # Random noise
+    0.0002 * np.random.randn(len(x_track))  # Random noise
 )
 
 sig_hmax = SpaceSignalProcessing(x_track, track_irregularity)
diff --git a/example_time_signal.py b/example_time_signal.py
index b0b9bbf..a707e90 100644
--- a/example_time_signal.py
+++ b/example_time_signal.py
@@ -2,14 +2,12 @@
 import matplotlib.pylab as plt
 from SignalProcessingTools.time_signal import TimeSignalProcessing, IntegrationRules, Windows
 
-
 # Create test data
 x = np.linspace(0, 100, 50001)
 omega = 2 * np.pi * 6
 y = 1.75 * np.sin(omega * x)
 y_noise = y + 0.01 * np.sin(120 * x)
 
-
 # Create a SignalProcessing object and demonstrate basic functionality
 print("------------------------------------")
 print("EXAMPLE 1: Basic FFT and integration")
@@ -47,12 +45,14 @@
 sig.reset()
 print(sig)
 
-
 # Example 2: Working with windowed signals and PSD
 print("---------------------------------------")
 print("EXAMPLE 2: Windowed processing and PSD")
 print("---------------------------------------")
-sig_window = TimeSignalProcessing(x, y_noise, window=Windows.HAMMING, window_size=4096)
+sig_window = TimeSignalProcessing(x,
+                                  y_noise,
+                                  window=Windows.HAMMING,
+                                  window_size=4096)
 print(sig_window)
 
 # Calculate and plot PSD
@@ -69,8 +69,10 @@
 # Calculate and plot spectrogram
 sig_window.spectrogram()
 plt.figure(figsize=(10, 6))
-plt.pcolormesh(sig_window.time_Sxx, sig_window.frequency_Sxx,
-               10 * np.log10(sig_window.Sxx), shading='gouraud')
+plt.pcolormesh(sig_window.time_Sxx,
+               sig_window.frequency_Sxx,
+               10 * np.log10(sig_window.Sxx),
+               shading='gouraud')
 plt.ylim(0, 20)
 plt.ylabel('Frequency [Hz]')
 plt.xlabel('Time [s]')
@@ -80,7 +82,6 @@
 
 print(sig_window)
 
-
 # Example 3: Signal filtering
 print("---------------------------")
 print("EXAMPLE 3: Signal filtering")
@@ -102,7 +103,9 @@
 sig_filter.filter(10, 4, type_filter="lowpass")
 
 plt.figure(figsize=(10, 6))
-plt.plot(sig_filter.time[:500], sig_filter.signal[:500], label='Filtered signal')
+plt.plot(sig_filter.time[:500],
+         sig_filter.signal[:500],
+         label='Filtered signal')
 plt.plot(x[:500], y[:500], '--', label='Original clean signal')
 plt.xlabel('Time [s]')
 plt.ylabel('Amplitude')
@@ -113,7 +116,6 @@
 
 print(sig_filter)
 
-
 # Example 4: Inverse FFT
 print("----------------------")
 print("EXAMPLE 4: Inverse FFT")
@@ -137,7 +139,10 @@
 
 plt.figure(figsize=(10, 6))
 plt.plot(sig_ifft.time[:500], sig_ifft.signal[:500], label='Original signal')
-plt.plot(sig_ifft.time_inv[:500], sig_ifft.signal_inv[:500], '--', label='Reconstructed signal')
+plt.plot(sig_ifft.time_inv[:500],
+         sig_ifft.signal_inv[:500],
+         '--',
+         label='Reconstructed signal')
 plt.xlabel('Time [s]')
 plt.ylabel('Amplitude')
 plt.title('Signal Reconstruction with Inverse FFT')
@@ -146,11 +151,11 @@
 plt.show()
 
 # Calculate RMSE between original and reconstructed signals
-rmse = np.sqrt(np.sum((sig_ifft.signal[:500] - sig_ifft.signal_inv[:500]) ** 2) / 500)
+rmse = np.sqrt(
+    np.sum((sig_ifft.signal[:500] - sig_ifft.signal_inv[:500])**2) / 500)
 print(f"RMSE between original and reconstructed signals: {rmse:.6f}")
 print(sig_ifft)
 
-
 # Example 5: Effective velocity calculation using SBR method
 print("-------------------------------------------")
 print("EXAMPLE 5: Effective velocity (SBR method)")
@@ -158,7 +163,8 @@
 
 # Create a vibration signal (using more complex multi-frequency signal)
 t = np.linspace(0, 10, 5001)
-vib_signal = 0.5 * np.sin(2 * np.pi * 2 * t) + 0.3 * np.sin(2 * np.pi * 8 * t) + 0.2 * np.sin(2 * np.pi * 15 * t)
+vib_signal = 0.5 * np.sin(2 * np.pi * 2 * t) + 0.3 * np.sin(
+    2 * np.pi * 8 * t) + 0.2 * np.sin(2 * np.pi * 15 * t)
 
 sig_veff = TimeSignalProcessing(t, vib_signal)
 # Calculate effective velocity
@@ -176,7 +182,6 @@
 
 print(sig_veff)
 
-
 # Example 6: Different integration rules
 print("---------------------------------------")
 print("EXAMPLE 6: Integration rules comparison")
@@ -196,7 +201,8 @@
 
 # Analytical integration for comparison (integral of sin(2πt) is -cos(2πt)/(2π))
 vel_analytical = -np.cos(2 * np.pi * 1 * t_acc) / (2 * np.pi)
-vel_analytical = vel_analytical - np.mean(vel_analytical)  # baseline correction for comparison
+vel_analytical = vel_analytical - np.mean(
+    vel_analytical)  # baseline correction for comparison
 
 plt.figure(figsize=(10, 6))
 plt.plot(t_acc, vel_analytical, 'k-', label='Analytical solution')
@@ -212,4 +218,4 @@
 print("Trapezoid integration:")
 print(sig_trap)
 print("\nSimpson integration:")
-print(sig_simp)
\ No newline at end of file
+print(sig_simp)
diff --git a/tests/test_space_signal.py b/tests/test_space_signal.py
index 732f430..5d8b8fc 100644
--- a/tests/test_space_signal.py
+++ b/tests/test_space_signal.py
@@ -32,12 +32,16 @@ def test_track_quality_index(test_data):
     with open("./tests/data/track_alignment_results.txt", "r") as fi:
         data = json.load(fi)
 
-    assert np.allclose(sig.coordinates, data["coordinates"], rtol=1e-5, atol=1e-8)
+    assert np.allclose(sig.coordinates,
+                       data["coordinates"],
+                       rtol=1e-5,
+                       atol=1e-8)
     assert np.allclose(sig.d0, data["D0"], rtol=1e-5, atol=1e-8)
     assert np.allclose(sig.d1, data["D1"], rtol=1e-5, atol=1e-8)
     assert np.allclose(sig.d2, data["D2"], rtol=1e-5, atol=1e-8)
     assert np.allclose(sig.d3, data["D3"], rtol=1e-5, atol=1e-8)
 
+
 def test_Hmax(test_data):
     """
     Test the Hmax function
@@ -48,19 +52,22 @@ def test_Hmax(test_data):
 
     sig.compute_Hmax()
 
-
-    rms_band_matlab = np.array([2087.55705457531, 1139.29553877343, 793.047457091564, 548.095561097181,
-                                648.015015656438, 521.608760233929, 790.563948013129, 886.097705321285,
-                                1342.44544888507])
-    h_max_matlab = np.array([6557.20346546820, 3764.49040821894, 2505.54201229720, 1727.21064286318,
-                                1208.73808675389, 1182.77849807824, 1996.50964604940, 2682.54554936051,
-                                3681.41028314498])
-    h_max_dx_matlab = np.array([293.263231128877, 783.641358934000, 1300.63422139907, 524.680132894043,
-                                139.978582308885, 583.726452878631, 662.410999843909, 1240.10908108192,
-                                1127.56942541745])
-
-
+    rms_band_matlab = np.array([
+        2087.55705457531, 1139.29553877343, 793.047457091564, 548.095561097181,
+        648.015015656438, 521.608760233929, 790.563948013129, 886.097705321285,
+        1342.44544888507
+    ])
+    h_max_matlab = np.array([
+        6557.20346546820, 3764.49040821894, 2505.54201229720, 1727.21064286318,
+        1208.73808675389, 1182.77849807824, 1996.50964604940, 2682.54554936051,
+        3681.41028314498
+    ])
+    h_max_dx_matlab = np.array([
+        293.263231128877, 783.641358934000, 1300.63422139907, 524.680132894043,
+        139.978582308885, 583.726452878631, 662.410999843909, 1240.10908108192,
+        1127.56942541745
+    ])
 
     assert np.allclose(sig.rms_bands, rms_band_matlab, rtol=1e-3, atol=1e-8)
     assert np.allclose(sig.max_fast, h_max_matlab, rtol=1e-3, atol=1e-8)
-    assert np.allclose(sig.max_fast_Dx, h_max_dx_matlab, rtol=1e-3, atol=1e-8)
\ No newline at end of file
+    assert np.allclose(sig.max_fast_Dx, h_max_dx_matlab, rtol=1e-3, atol=1e-8)
diff --git a/tests/test_time_signal.py b/tests/test_time_signal.py
index b9bbe90..60a649b 100644
--- a/tests/test_time_signal.py
+++ b/tests/test_time_signal.py
@@ -8,6 +8,7 @@
 FREQ = 6
 AMP = 1.75
 
+
 @pytest.fixture
 def test_data():
     """
@@ -19,6 +20,7 @@ def test_data():
     y_noise = y + 0.01 * np.sin(120 * x)
     return x, y, y_noise
 
+
 def test_fft(test_data):
     """
     Test the fft function
@@ -33,10 +35,10 @@ def test_fft(test_data):
     assert len(sig.signal) == 50001
     assert len(sig.time) == 50001
 
-    np.testing.assert_almost_equal(sig.frequency[np.argmax(sig.amplitude)], FREQ, 2)
+    np.testing.assert_almost_equal(sig.frequency[np.argmax(sig.amplitude)],
+                                   FREQ, 2)
     np.testing.assert_almost_equal(np.max(sig.amplitude), AMP, 2)
 
-
     # results full representation
     sig.fft(half_representation=False)
 
@@ -44,16 +46,23 @@ def test_fft(test_data):
     assert len(sig.signal) == 50001
     assert len(sig.time) == 50001
 
-    np.testing.assert_almost_equal(sig.frequency[np.argmax(sig.amplitude[:int(len(sig.amplitude) / 2)])], FREQ, 2)
-    np.testing.assert_almost_equal(np.max(sig.amplitude[:int(len(sig.amplitude) / 2)]), AMP / 2, 2)
+    np.testing.assert_almost_equal(
+        sig.frequency[np.argmax(sig.amplitude[:int(len(sig.amplitude) / 2)])],
+        FREQ, 2)
+    np.testing.assert_almost_equal(
+        np.max(sig.amplitude[:int(len(sig.amplitude) / 2)]), AMP / 2, 2)
 
     # example with spectral leakage
     y = 1.75 * np.sin(2.675 * 2 * np.pi * x)
     sig = TimeSignalProcessing(x, y)
     sig.fft()
 
-    np.testing.assert_almost_equal(sig.frequency[np.argmax(sig.amplitude[:int(len(sig.amplitude) / 2)])], 2.675, 2)
-    np.testing.assert_almost_equal(np.max(sig.amplitude[:int(len(sig.amplitude) / 2)]), 1.137, 2)
+    np.testing.assert_almost_equal(
+        sig.frequency[np.argmax(sig.amplitude[:int(len(sig.amplitude) / 2)])],
+        2.675, 2)
+    np.testing.assert_almost_equal(
+        np.max(sig.amplitude[:int(len(sig.amplitude) / 2)]), 1.137, 2)
+
 
 def test_fft_nb_points(test_data):
     """
@@ -66,10 +75,11 @@ def test_fft_nb_points(test_data):
     sig.fft(nb_points=2**18)
 
     # check if signal lenght has been adapted to window size
-    assert len(sig.amplitude) == (2**18)/2
-    assert len(sig.frequency) == (2**18)/2
+    assert len(sig.amplitude) == (2**18) / 2
+    assert len(sig.frequency) == (2**18) / 2
 
-    np.testing.assert_almost_equal(sig.frequency[np.argmax(sig.amplitude)], FREQ, 2)
+    np.testing.assert_almost_equal(sig.frequency[np.argmax(sig.amplitude)],
+                                   FREQ, 2)
     np.testing.assert_almost_equal(np.max(sig.amplitude), AMP, 2)
 
     # results full representation
@@ -79,8 +89,11 @@ def test_fft_nb_points(test_data):
     assert len(sig.amplitude) == 2**18
     assert len(sig.frequency) == 2**18
 
-    np.testing.assert_almost_equal(sig.frequency[np.argmax(sig.amplitude[:int(len(sig.amplitude) / 2)])], FREQ, 2)
-    np.testing.assert_almost_equal(np.max(sig.amplitude[:int(len(sig.amplitude) / 2)]), AMP / 2, 2)
+    np.testing.assert_almost_equal(
+        sig.frequency[np.argmax(sig.amplitude[:int(len(sig.amplitude) / 2)])],
+        FREQ, 2)
+    np.testing.assert_almost_equal(
+        np.max(sig.amplitude[:int(len(sig.amplitude) / 2)]), AMP / 2, 2)
 
 
 def test_fft_window(test_data):
@@ -90,7 +103,9 @@ def test_fft_window(test_data):
     x, y, _ = test_data
 
     # assert that sig raises a Value error
-    with pytest.raises(ValueError, match="When using a window the `window_size` must be specified"):
+    with pytest.raises(
+            ValueError,
+            match="When using a window the `window_size` must be specified"):
         sig = TimeSignalProcessing(x, y, window=Windows.HAMMING)
 
     # test with window - half representation
@@ -101,8 +116,11 @@ def test_fft_window(test_data):
     assert len(sig.signal) == 54000
     assert len(sig.time) == 54000
 
-    np.testing.assert_almost_equal(sig.frequency[np.argmax(sig.amplitude[:int(len(sig.amplitude))])], FREQ, 2)
-    np.testing.assert_almost_equal(np.max(sig.amplitude[:int(len(sig.amplitude))]), AMP, 2)
+    np.testing.assert_almost_equal(
+        sig.frequency[np.argmax(sig.amplitude[:int(len(sig.amplitude))])],
+        FREQ, 2)
+    np.testing.assert_almost_equal(
+        np.max(sig.amplitude[:int(len(sig.amplitude))]), AMP, 2)
 
     # full representation
     sig.fft(half_representation=False)
@@ -111,8 +129,11 @@ def test_fft_window(test_data):
     assert len(sig.signal) == 54000
     assert len(sig.time) == 54000
 
-    np.testing.assert_almost_equal(sig.frequency[np.argmax(sig.amplitude[:int(len(sig.amplitude))])], FREQ, 2)
-    np.testing.assert_almost_equal(np.max(sig.amplitude[:int(len(sig.amplitude))]), AMP / 2, 2)
+    np.testing.assert_almost_equal(
+        sig.frequency[np.argmax(sig.amplitude[:int(len(sig.amplitude))])],
+        FREQ, 2)
+    np.testing.assert_almost_equal(
+        np.max(sig.amplitude[:int(len(sig.amplitude))]), AMP / 2, 2)
 
 
 def test_ifft(test_data):
@@ -126,7 +147,11 @@ def test_ifft(test_data):
     sig.fft(half_representation=True)
 
     # assert that sig raises a Value error
-    with pytest.raises(NotImplementedError, match="Half representation not supported for inverse FFT. Please compute FFT with full representation."):
+    with pytest.raises(
+            NotImplementedError,
+            match=
+            "Half representation not supported for inverse FFT. Please compute FFT with full representation."
+    ):
         sig.inv_fft()
 
     # test with window - full representation
@@ -138,8 +163,9 @@ def test_ifft(test_data):
     # check if signal lenght has been adapted to window size
     assert len(sig.signal) == len(sig.signal_inv)
 
-    rmse = np.sqrt(np.sum((sig.signal - sig.signal_inv) ** 2) / len(y))
-    assert(rmse < TOL)
+    rmse = np.sqrt(np.sum((sig.signal - sig.signal_inv)**2) / len(y))
+    assert (rmse < TOL)
+
 
 def test_ifft_window(test_data):
     """
@@ -152,7 +178,11 @@ def test_ifft_window(test_data):
     sig.fft(half_representation=True)
 
     # assert that sig raises a Value error
-    with pytest.raises(NotImplementedError, match="Half representation not supported for inverse FFT. Please compute FFT with full representation."):
+    with pytest.raises(
+            NotImplementedError,
+            match=
+            "Half representation not supported for inverse FFT. Please compute FFT with full representation."
+    ):
         sig.inv_fft()
 
     # test with window - half representation
@@ -160,7 +190,9 @@ def test_ifft_window(test_data):
     sig.fft(half_representation=False)
 
     # assert that sig raises a Value error
-    with pytest.raises(ValueError, match="Cannot perform inverse FFT on the windowed signal."):
+    with pytest.raises(
+            ValueError,
+            match="Cannot perform inverse FFT on the windowed signal."):
         sig.inv_fft()
 
 
@@ -175,13 +207,18 @@ def test_int(test_data):
     omega = 2 * np.pi * FREQ
     int_sig = -AMP * np.cos(omega * x) / omega
 
-    rmse = np.sqrt(np.sum((sig.signal - int_sig) ** 2) / len(int_sig))
-    assert(rmse < TOL)
+    rmse = np.sqrt(np.sum((sig.signal - int_sig)**2) / len(int_sig))
+    assert (rmse < TOL)
 
     sig = TimeSignalProcessing(x, y)
-    sig.integrate(baseline=True, hp=True, rule=IntegrationRules.SIMPSON, fpass=1, n=6)
-    rmse = np.sqrt(np.sum((sig.signal - int_sig) ** 2) / len(int_sig))
-    assert(rmse < TOL)
+    sig.integrate(baseline=True,
+                  hp=True,
+                  rule=IntegrationRules.SIMPSON,
+                  fpass=1,
+                  n=6)
+    rmse = np.sqrt(np.sum((sig.signal - int_sig)**2) / len(int_sig))
+    assert (rmse < TOL)
+
 
 def test_filter(test_data):
     """
@@ -192,15 +229,21 @@ def test_filter(test_data):
     sig.filter(10, 4, type_filter="lowpass")
 
     # compare between 200 and -200 to avoid edge effects
-    rmse = np.sqrt(np.sum((sig.signal[200:-200] - y[200:-200]) ** 2) / len(y[200:-200]))
-    assert(rmse < TOL)
+    rmse = np.sqrt(
+        np.sum((sig.signal[200:-200] - y[200:-200])**2) / len(y[200:-200]))
+    assert (rmse < TOL)
+
 
 def test_psd(test_data):
     """
     Test the psd function
     """
     x, y, _ = test_data
-    with pytest.raises(ValueError, match="No window defined. Please define a window when initialising SignalProcessing."):
+    with pytest.raises(
+            ValueError,
+            match=
+            "No window defined. Please define a window when initialising SignalProcessing."
+    ):
         sig = TimeSignalProcessing(x, y)
         sig.psd()
 
@@ -208,17 +251,18 @@ def test_psd(test_data):
     sig.psd()
 
     # power
-    power_sinus_wave = AMP ** 2 / 2
+    power_sinus_wave = AMP**2 / 2
     bin_width = sig.Fs / sig.window_size
-    ENBW = np.sum(sig.window ** 2) / (np.sum(sig.window)**2) * sig.window_size
+    ENBW = np.sum(sig.window**2) / (np.sum(sig.window)**2) * sig.window_size
     peak_psd = power_sinus_wave / (ENBW * bin_width)
 
-    np.testing.assert_almost_equal(sig.frequency_Pxx[np.argmax(sig.Pxx)], FREQ, 2)
+    np.testing.assert_almost_equal(sig.frequency_Pxx[np.argmax(sig.Pxx)], FREQ,
+                                   2)
     assert (np.abs((np.max(sig.Pxx) - peak_psd) / peak_psd) < 0.035)
 
     # test the time interpolation
-    assert len(sig.time) == np.ceil(50001/4000)*4000
-    assert ((np.diff(sig.time)-1/500) < FLOAT_TOL).all()
+    assert len(sig.time) == np.ceil(50001 / 4000) * 4000
+    assert ((np.diff(sig.time) - 1 / 500) < FLOAT_TOL).all()
 
 
 def test_v_eff():
@@ -235,7 +279,7 @@ def test_v_eff():
         v_eff = fi.read().splitlines()
     v_eff = np.array([list(map(float, r.split(";"))) for r in v_eff])
 
-    time = np.linspace(0, (raw.shape[0]-1) / 500, raw.shape[0])
+    time = np.linspace(0, (raw.shape[0] - 1) / 500, raw.shape[0])
 
     # compute veff
     for i in range(raw.shape[1]):
@@ -243,6 +287,7 @@ def test_v_eff():
         sig.v_eff_SBR()
         np.testing.assert_almost_equal(sig.v_eff, np.array(v_eff)[:, i], 2)
 
+
 def test_str_representation(test_data):
     """
     Test the __str__ method to verify operations are tracked correctly
@@ -283,6 +328,7 @@ def test_str_representation(test_data):
     assert any("Filter" in op for op in sig.operations)
     assert any("PSD" in op for op in sig.operations)
 
+
 def test_spectrogram(test_data):
     """
     Test the spectrogram function
@@ -293,10 +339,11 @@ def test_spectrogram(test_data):
 
     # check if signal lenght has been adapted to window size
     assert sig.Sxx.shape == (301, 95)
-    assert sig.time_Sxx.shape == (95,)
-    assert sig.frequency_Sxx.shape == (301,)
+    assert sig.time_Sxx.shape == (95, )
+    assert sig.frequency_Sxx.shape == (301, )
 
-    np.testing.assert_almost_equal(sig.frequency_Sxx[np.where(sig.Sxx==np.max(sig.Sxx))[0][0]], FREQ, 0)
+    np.testing.assert_almost_equal(
+        sig.frequency_Sxx[np.where(sig.Sxx == np.max(sig.Sxx))[0][0]], FREQ, 0)
     np.testing.assert_almost_equal(np.max(sig.Sxx), 1.27, 2)
 
     # # plot spectrogram
@@ -355,7 +402,10 @@ def test_reset(test_data):
     assert sig.time_Sxx is None
 
     # Verify FFT settings are reset
-    assert sig.fft_settings == {"nb_points": None, "half_representation": False}
+    assert sig.fft_settings == {
+        "nb_points": None,
+        "half_representation": False
+    }
 
     # Verify operations history is cleared
     assert len(sig.operations) == 0
@@ -363,6 +413,7 @@ def test_reset(test_data):
     # Verify string representation shows no operations
     assert "No operations performed yet" in str(sig)
 
+
 def test_one_third_octave(test_data):
     """
     Test the one third octave function
@@ -370,11 +421,13 @@ def test_one_third_octave(test_data):
     x, y, _ = test_data
 
     # definition of frequencies used in the test
-    freqs_used = [10, 12.5, 16, 20, 25, 31.5, 40, 50, 63, 80, 100, 125, 160, 200, 250]
+    freqs_used = [
+        10, 12.5, 16, 20, 25, 31.5, 40, 50, 63, 80, 100, 125, 160, 200, 250
+    ]
     # compute the power at 20 Hz
-    f_center = 1000 * (2 ** ((-17) / 3))
-    f_max = f_center * (2 ** (1 / 6))
-    f_min = f_center / (2 ** (1 / 6))
+    f_center = 1000 * (2**((-17) / 3))
+    f_max = f_center * (2**(1 / 6))
+    f_min = f_center / (2**(1 / 6))
 
     # test FFT
     sig = TimeSignalProcessing(x, y)
@@ -385,7 +438,7 @@ def test_one_third_octave(test_data):
 
     idx = np.where((sig.frequency >= f_min) & (sig.frequency < f_max))[0]
 
-    assert sig.octave_bands_fft_power[3] == np.sum(sig.amplitude[idx] ** 2)
+    assert sig.octave_bands_fft_power[3] == np.sum(sig.amplitude[idx]**2)
     assert sig.octave_bands_fft[3] == 20
 
     # test PSDF
@@ -394,7 +447,8 @@ def test_one_third_octave(test_data):
     sig.one_third_octave_bands()
     assert all(sig.octave_bands_Pxx == freqs_used)
 
-    idx = np.where((sig.frequency_Pxx >= f_min) & (sig.frequency_Pxx < f_max))[0]
+    idx = np.where((sig.frequency_Pxx >= f_min)
+                   & (sig.frequency_Pxx < f_max))[0]
     delta_freq = sig.frequency_Pxx[1] - sig.frequency_Pxx[0]
     assert sig.octave_bands_Pxx_power[3] == np.sum(sig.Pxx[idx] * delta_freq)
     assert sig.octave_bands_Pxx[3] == 20