Laplace Transform
--------------------------------------------------------------------------- ModuleNotFoundError Traceback (most recent call last) File /tmp/ipykernel_22249/705522950.py:2 1 #%matplotlib notebook ----> 2 get_ipython().run_line_magic('matplotlib', 'inline') 4 from matplotlib import patches 6 import numpy as np File /media/anwaldt/ANWALDT_DATA/WORK/TEACHING/Online/RingBuffer/py-env/lib/python3.12/site-packages/IPython/core/interactiveshell.py:2480, in InteractiveShell.run_line_magic(self, magic_name, line, _stack_depth) 2478 kwargs['local_ns'] = self.get_local_scope(stack_depth) 2479 with self.builtin_trap: -> 2480 result = fn(*args, **kwargs) 2482 # The code below prevents the output from being displayed 2483 # when using magics with decorator @output_can_be_silenced 2484 # when the last Python token in the expression is a ';'. 2485 if getattr(fn, magic.MAGIC_OUTPUT_CAN_BE_SILENCED, False): File /media/anwaldt/ANWALDT_DATA/WORK/TEACHING/Online/RingBuffer/py-env/lib/python3.12/site-packages/IPython/core/magics/pylab.py:103, in PylabMagics.matplotlib(self, line) 98 print( 99 "Available matplotlib backends: %s" 100 % _list_matplotlib_backends_and_gui_loops() 101 ) 102 else: --> 103 gui, backend = self.shell.enable_matplotlib(args.gui) 104 self._show_matplotlib_backend(args.gui, backend) File /media/anwaldt/ANWALDT_DATA/WORK/TEACHING/Online/RingBuffer/py-env/lib/python3.12/site-packages/IPython/core/interactiveshell.py:3665, in InteractiveShell.enable_matplotlib(self, gui) 3662 import matplotlib_inline.backend_inline 3664 from IPython.core import pylabtools as pt -> 3665 gui, backend = pt.find_gui_and_backend(gui, self.pylab_gui_select) 3667 if gui != None: 3668 # If we have our first gui selection, store it 3669 if self.pylab_gui_select is None: File /media/anwaldt/ANWALDT_DATA/WORK/TEACHING/Online/RingBuffer/py-env/lib/python3.12/site-packages/IPython/core/pylabtools.py:338, in find_gui_and_backend(gui, gui_select) 321 def find_gui_and_backend(gui=None, gui_select=None): 322 """Given a gui string return the gui and mpl backend. 323 324 Parameters (...) 335 'WXAgg','Qt4Agg','module://matplotlib_inline.backend_inline','agg'). 336 """ --> 338 import matplotlib 340 if _matplotlib_manages_backends(): 341 backend_registry = matplotlib.backends.registry.backend_registry ModuleNotFoundError: No module named 'matplotlib'
--------------------------------------------------------------------------- ModuleNotFoundError Traceback (most recent call last) Cell In[2], line 1 ----> 1 get_ipython().run_line_magic('run', 'header.ipynb') File /media/anwaldt/ANWALDT_DATA/WORK/TEACHING/Online/RingBuffer/py-env/lib/python3.12/site-packages/IPython/core/interactiveshell.py:2480, in InteractiveShell.run_line_magic(self, magic_name, line, _stack_depth) 2478 kwargs['local_ns'] = self.get_local_scope(stack_depth) 2479 with self.builtin_trap: -> 2480 result = fn(*args, **kwargs) 2482 # The code below prevents the output from being displayed 2483 # when using magics with decorator @output_can_be_silenced 2484 # when the last Python token in the expression is a ';'. 2485 if getattr(fn, magic.MAGIC_OUTPUT_CAN_BE_SILENCED, False): File /media/anwaldt/ANWALDT_DATA/WORK/TEACHING/Online/RingBuffer/py-env/lib/python3.12/site-packages/IPython/core/magics/execution.py:741, in ExecutionMagics.run(self, parameter_s, runner, file_finder) 739 with preserve_keys(self.shell.user_ns, '__file__'): 740 self.shell.user_ns['__file__'] = filename --> 741 self.shell.safe_execfile_ipy(filename, raise_exceptions=True) 742 return 744 # Control the response to exit() calls made by the script being run File /media/anwaldt/ANWALDT_DATA/WORK/TEACHING/Online/RingBuffer/py-env/lib/python3.12/site-packages/IPython/core/interactiveshell.py:3005, in InteractiveShell.safe_execfile_ipy(self, fname, shell_futures, raise_exceptions) 3003 result = self.run_cell(cell, silent=True, shell_futures=shell_futures) 3004 if raise_exceptions: -> 3005 result.raise_error() 3006 elif not result.success: 3007 break File /media/anwaldt/ANWALDT_DATA/WORK/TEACHING/Online/RingBuffer/py-env/lib/python3.12/site-packages/IPython/core/interactiveshell.py:308, in ExecutionResult.raise_error(self) 306 raise self.error_before_exec 307 if self.error_in_exec is not None: --> 308 raise self.error_in_exec [... skipping hidden 1 frame] File /tmp/ipykernel_22249/705522950.py:2 1 #%matplotlib notebook ----> 2 get_ipython().run_line_magic('matplotlib', 'inline') 4 from matplotlib import patches 6 import numpy as np File /media/anwaldt/ANWALDT_DATA/WORK/TEACHING/Online/RingBuffer/py-env/lib/python3.12/site-packages/IPython/core/interactiveshell.py:2480, in InteractiveShell.run_line_magic(self, magic_name, line, _stack_depth) 2478 kwargs['local_ns'] = self.get_local_scope(stack_depth) 2479 with self.builtin_trap: -> 2480 result = fn(*args, **kwargs) 2482 # The code below prevents the output from being displayed 2483 # when using magics with decorator @output_can_be_silenced 2484 # when the last Python token in the expression is a ';'. 2485 if getattr(fn, magic.MAGIC_OUTPUT_CAN_BE_SILENCED, False): File /media/anwaldt/ANWALDT_DATA/WORK/TEACHING/Online/RingBuffer/py-env/lib/python3.12/site-packages/IPython/core/magics/pylab.py:103, in PylabMagics.matplotlib(self, line) 98 print( 99 "Available matplotlib backends: %s" 100 % _list_matplotlib_backends_and_gui_loops() 101 ) 102 else: --> 103 gui, backend = self.shell.enable_matplotlib(args.gui) 104 self._show_matplotlib_backend(args.gui, backend) File /media/anwaldt/ANWALDT_DATA/WORK/TEACHING/Online/RingBuffer/py-env/lib/python3.12/site-packages/IPython/core/interactiveshell.py:3665, in InteractiveShell.enable_matplotlib(self, gui) 3662 import matplotlib_inline.backend_inline 3664 from IPython.core import pylabtools as pt -> 3665 gui, backend = pt.find_gui_and_backend(gui, self.pylab_gui_select) 3667 if gui != None: 3668 # If we have our first gui selection, store it 3669 if self.pylab_gui_select is None: File /media/anwaldt/ANWALDT_DATA/WORK/TEACHING/Online/RingBuffer/py-env/lib/python3.12/site-packages/IPython/core/pylabtools.py:338, in find_gui_and_backend(gui, gui_select) 321 def find_gui_and_backend(gui=None, gui_select=None): 322 """Given a gui string return the gui and mpl backend. 323 324 Parameters (...) 335 'WXAgg','Qt4Agg','module://matplotlib_inline.backend_inline','agg'). 336 """ --> 338 import matplotlib 340 if _matplotlib_manages_backends(): 341 backend_registry = matplotlib.backends.registry.backend_registry ModuleNotFoundError: No module named 'matplotlib'
The Laplace transform is a time frequency transform allowing the analysis of LTI systems in terms of their impulse response behavior. This is relevant when designing and investigating filters or other processing units. The transform is defined by the following integral:
$$ F(s) = \int\limits_0^\infty e^{-s t} f(t) dt $$
$s$ is the Laplace operator
$$ s = \sigma + j \omega $$
which can be decomposed into two terms:
$$ F(s) = \int\limits_0^\infty e^{-\sigma t} \ e^{-j \omega t} f(t) dt $$
The term $e^{-\sigma t}$ is referred to as the decay term, whereas $ e^{-j \omega t}$ will be called the frequency term.
<>:29: SyntaxWarning: invalid escape sequence '\o'
<>:47: SyntaxWarning: invalid escape sequence '\s'
<>:29: SyntaxWarning: invalid escape sequence '\o'
<>:47: SyntaxWarning: invalid escape sequence '\s'
/tmp/ipykernel_22159/1301335065.py:29: SyntaxWarning: invalid escape sequence '\o'
plt.ylabel('$\omega$');
/tmp/ipykernel_22159/1301335065.py:47: SyntaxWarning: invalid escape sequence '\s'
plt.ylabel('$\sigma$');
/tmp/ipykernel_22159/1301335065.py:29: SyntaxWarning: invalid escape sequence '\o'
plt.ylabel('$\omega$');
/tmp/ipykernel_22159/1301335065.py:47: SyntaxWarning: invalid escape sequence '\s'
plt.ylabel('$\sigma$');
--------------------------------------------------------------------------- NameError Traceback (most recent call last) Cell In[2], line 16 11 out[i] = np.imag(np.exp(-s*val) * np.exp(-1j * w * val)) 13 return out ---> 16 plt.figure() 18 t = np.linspace(0,1,1000) 20 for i in range(10): NameError: name 'plt' is not defined