Merge pull request #4 from FOSSEE/master

Update fork. DONE

7 files changed, 355 insertions, 0 deletions

diff --git a/FSF-2020/calculus/series-and-transformations/Z-Transform/README.md b/FSF-2020/calculus/series-and-transformations/Z-Transform/README.md
new file mode 100644
index 0000000..c626bdf
--- /dev/null
+++ b/FSF-2020/calculus/series-and-transformations/Z-Transform/README.md
@@ -0,0 +1,9 @@
+#### Sampling
+![GIF1](gifs/file1.gif)
+
+#### Z Transform of a delta function
+![GIF2](gifs/file2.gif)
+
+#### Region of convergence
+![GIF3](gifs/file3.gif)
+
diff --git a/FSF-2020/calculus/series-and-transformations/Z-Transform/gifs/file1.gif b/FSF-2020/calculus/series-and-transformations/Z-Transform/gifs/file1.gif
new file mode 100644
index 0000000..d21aa59
--- /dev/null
+++ b/FSF-2020/calculus/series-and-transformations/Z-Transform/gifs/file1.gif
diff --git a/FSF-2020/calculus/series-and-transformations/Z-Transform/gifs/file2.gif b/FSF-2020/calculus/series-and-transformations/Z-Transform/gifs/file2.gif
new file mode 100644
index 0000000..203be8d
--- /dev/null
+++ b/FSF-2020/calculus/series-and-transformations/Z-Transform/gifs/file2.gif
diff --git a/FSF-2020/calculus/series-and-transformations/Z-Transform/gifs/file3.gif b/FSF-2020/calculus/series-and-transformations/Z-Transform/gifs/file3.gif
new file mode 100644
index 0000000..0f100f1
--- /dev/null
+++ b/FSF-2020/calculus/series-and-transformations/Z-Transform/gifs/file3.gif
diff --git a/FSF-2020/calculus/series-and-transformations/Z-Transform/video1_Sampling.py b/FSF-2020/calculus/series-and-transformations/Z-Transform/video1_Sampling.py
new file mode 100644
index 0000000..47615e3
--- /dev/null
+++ b/FSF-2020/calculus/series-and-transformations/Z-Transform/video1_Sampling.py
@@ -0,0 +1,81 @@
+from manimlib.imports import *
+import math
+
+def func(x):
+    return math.pow(x,3)-2*math.pow(x,2)-x+3
+
+class graphScene(GraphScene):
+    CONFIG = {
+        "x_min": -3,
+        "x_max": 3,
+        "y_min": -4,
+        "y_max": 4,
+        "x_tick_frequency": 0.2,
+        "graph_origin": ORIGIN,
+        "function_color": RED,
+        "axes_color": BLUE,
+        "x_axis_label": "$t$",
+        "y_axis_label": "$f(t)$",
+        "exclude_zero_label": True,
+        "x_labeled_nums": range(-3, 4, 1),
+        "y_axis_height": 5,
+        "x_axis_width": 9,
+    }
+
+    def construct(self):
+        x_each_unit = self.x_axis_width / (self.x_max - self.x_min)
+        y_each_unit = self.y_axis_height / (self.y_max - self.y_min)   
+
+        fx=TextMobject("$f(t) = { t }^{ 3 }{ -2t }^{ 2 }-t+3$").set_color(RED).to_corner(UP+RIGHT).scale(0.4)
+        self.setup_axes(animate=True,scalee=1)
+        function=self.get_graph(lambda x:math.pow(x,3)-2*math.pow(x,2)-x+3,color=RED,x_min=-1,x_max=2)
+        functionArea=self.get_riemann_rectangles(function,x_min=-1,x_max=2,dx=0.01,start_color=GREEN,end_color=YELLOW,stroke_color=GREEN,fill_opacity=0.8)
+        functionDot=Dot(point=self.graph_origin,radius=0.065,color=WHITE)
+        aboveText1=TextMobject("Continuous","Time Function").shift(4*RIGHT+2*UP).scale(0.4).set_color_by_tex_to_color_map({"Continuous":YELLOW,"Time Function":BLUE})
+        aboveText2=TextMobject("Discrete","Time Function").shift(4*RIGHT+2*UP).scale(0.4).set_color_by_tex_to_color_map({"Time Function":BLUE,"Discrete":YELLOW})
+
+        bottomText1=TextMobject("Instead of considering the","function","over the","entire $t$,").shift(4.5*RIGHT+3*DOWN).scale(0.4).set_color_by_tex_to_color_map({"entire $t$,":RED,"function":YELLOW})
+        bottomText2=TextMobject("We consider only at","certain $t$").shift(4.5*RIGHT+3*DOWN).scale(0.4).set_color_by_tex_to_color_map({"certain $t$":RED})
+        
+        self.play(ShowCreation(function),Write(fx),FadeIn(aboveText1))
+        self.wait(0.7)
+        self.play(Write(bottomText1))
+        self.play(ShowCreation(functionArea),MoveAlongPath(functionDot,function))
+        self.wait(0.7)
+        self.play(FadeOut(bottomText1))
+        self.play(Write(bottomText2),FadeOut(aboveText1))
+        
+        dots=[Dot(radius=0.05) for i in range(10)]
+        dotShifts=[-1,-0.7,-0.4,0,0.3,0.6,1,1.3,1.6,2]
+        lines=[]
+        for x in dotShifts:
+            lines.append(Line(start=(x*x_each_unit,func(x)*y_each_unit,0),end=(x*x_each_unit,0,0),color=GREEN))
+        for i in range(10):
+            dots[i].shift(ORIGIN+RIGHT*x_each_unit*dotShifts[i]+y_each_unit*UP*func(dotShifts[i]))
+        updatedGraph=VGroup(dots[0],
+                            dots[1],
+                            dots[2],
+                            dots[3],
+                            dots[4],
+                            dots[5],
+                            dots[6],
+                            dots[7],
+                            dots[8],
+                            dots[9])
+        updatedGraph1=VGroup(
+                            lines[0],
+                            lines[1],
+                            lines[2],
+                            lines[3],
+                            lines[4],
+                            lines[5],
+                            lines[6],
+                            lines[7],
+                            lines[8],
+                            lines[9])
+
+        self.play(FadeOut(functionDot))
+        self.play(FadeOut(function),FadeIn(updatedGraph))
+        self.play(FadeOut(functionArea),FadeIn(updatedGraph1))
+        self.play(FadeOut(bottomText2),FadeIn(aboveText2))
+        self.wait(2)
\ No newline at end of file
diff --git a/FSF-2020/calculus/series-and-transformations/Z-Transform/video2_ZTransformOfDelta.py b/FSF-2020/calculus/series-and-transformations/Z-Transform/video2_ZTransformOfDelta.py
new file mode 100644
index 0000000..3063aa6
--- /dev/null
+++ b/FSF-2020/calculus/series-and-transformations/Z-Transform/video2_ZTransformOfDelta.py
@@ -0,0 +1,121 @@
+from manimlib.imports import *
+import numpy as np
+import math
+
+class deltaTransformation(GraphScene):
+    CONFIG = {
+        "x_min": -3,
+        "x_max": 3,
+        "y_min": -5,
+        "y_max": 5,
+        "graph_origin": ORIGIN,
+        "function_color": RED,
+        "axes_color": BLUE,
+        "x_axis_label": "$t$",
+        "y_axis_label": "$f(t)$",
+        "x_labeled_nums": range(-3, 4, 1),
+        # "y_axis_height": 4,
+        # "x_axis_width": 6,
+    }
+    def construct(self):
+        x_each_unit = self.x_axis_width / (self.x_max - self.x_min)
+        y_each_unit = self.y_axis_height / (self.y_max - self.y_min)   
+        self.setup_axes(animate=True,scalee=0.8)  
+        function=TextMobject("$f(t) = 2{ \delta  }_{ 0 }(t)+3{ \delta  }_{ 1 }(t)+4{ \delta  }_{ 2 }(t)$").scale(0.4).shift(5*RIGHT+3*UP).set_color(RED)
+        self.play(FadeIn(function))
+        twoDGraph=[
+                    Line(start=(0,0,0),end=(0,2*y_each_unit,0),color=GREEN),
+                    Line(start=(1*x_each_unit,0,0),end=(x_each_unit,3*y_each_unit,0),color=GREEN),
+                    Line(start=(2*x_each_unit,0,0),end=(2*x_each_unit,4*y_each_unit,0),color=GREEN)
+        ]
+        groupGraph=VGroup(twoDGraph[1],twoDGraph[2],self.axes,twoDGraph[0])
+        self.play(Write(twoDGraph[0]),ShowCreation(twoDGraph[1]),ShowCreation(twoDGraph[2]))
+        self.wait(1.2)
+        self.play(ApplyMethod(groupGraph.scale,0.7))
+        self.play(ApplyMethod(groupGraph.shift,5*LEFT),ApplyMethod(function.move_to,5*LEFT+3*UP))
+        self.graph_origin=2*RIGHT+2.5*DOWN
+        self.x_axis_width=6
+        self.x_axis_label="$|z|$"
+        self.y_axis_label="$|F(t)|$"
+        self.x_min=-3
+        self.x_max=6
+        self.y_min=-1
+        self.y_max=7
+        self.x_labeled_nums=range(-3,7,1)
+        self.setup_axes(animate=True,scalee=0.6)
+        x_each_unit = self.x_axis_width / (self.x_max - self.x_min)
+        y_each_unit = self.y_axis_height / (self.y_max - self.y_min)   
+        rightSideGraphs=[
+                        self.get_graph(lambda x:2,x_min=0,x_max=6,color=GREEN),
+                        self.get_graph(lambda x:2+3/x,x_min=0.6,x_max=6,color=GREEN),
+                        self.get_graph(lambda x:2+(3/x)+(4/x**2),x_min=1.24,x_max=6,color=GREEN)
+        ]
+        graphCoeff=[
+                    TextMobject("$2$").scale(0.4).shift(self.graph_origin+x_each_unit*RIGHT*2+UP*y_each_unit*2+DOWN*y_each_unit*0.5).set_color(RED),
+                    TextMobject("$2+\\frac { 3 }{ |z| }$").scale(0.4).shift(self.graph_origin+x_each_unit*RIGHT*3+UP*y_each_unit*2).set_color(RED),
+                    TextMobject("$2+\\frac { 3 }{ |z| } +\\frac { 4 }{ { |z| }^{ 2 } } $").scale(0.4).shift(self.graph_origin+x_each_unit*RIGHT*3.5+UP*y_each_unit*2).set_color(RED)
+        ]
+        self.play(ReplacementTransform(twoDGraph[0],rightSideGraphs[0]),FadeIn(graphCoeff[0]))
+        self.wait(0.5)
+        self.play(FadeOut(rightSideGraphs[0]),ReplacementTransform(twoDGraph[1],rightSideGraphs[1]),ReplacementTransform(graphCoeff[0],graphCoeff[1]))
+        self.wait(0.5)
+        self.play(FadeOut(rightSideGraphs[1]),ReplacementTransform(twoDGraph[2],rightSideGraphs[2]),ReplacementTransform(graphCoeff[1],graphCoeff[2]))
+        
+        self.wait(2)
+
+
+class graphCont(GraphScene,MovingCameraScene):
+    CONFIG = {
+        "x_min": -3,
+        "x_max": 6,
+        "y_min": -1,
+        "y_max": 7,
+        "graph_origin": 2*RIGHT+2.5*DOWN,
+        "function_color": RED,
+        "axes_color": BLUE,
+        "x_axis_label": "$|z|$",
+        "y_axis_label": "$|F(t)|$",
+        "exclude_zero_label": True,
+        "x_labeled_nums": range(-3, 7, 1),
+        "x_axis_width": 6,
+    }
+    def setup(self):
+        GraphScene.setup(self)
+        MovingCameraScene.setup(self)
+
+    def construct(self):
+        x_each_unit = self.x_axis_width / (self.x_max - self.x_min)
+        y_each_unit = self.y_axis_height / (self.y_max - self.y_min)   
+
+        coeff=TextMobject("$2+\\frac { 3 }{ |z| } +\\frac { 4 }{ { |z| }^{ 2 } } $").scale(0.4).shift(self.graph_origin+x_each_unit*RIGHT*3.5+UP*y_each_unit*2).set_color(RED)
+        self.setup_axes(scalee=0.6)
+        graph=self.get_graph(lambda x:2+(3/x)+(4/x**2),x_min=1.24,x_max=6,color=GREEN)
+        xAxis=self.get_graph(lambda x:0,x_min=1.24,x_max=6).shift(3*LEFT)
+        self.add(graph)
+        self.add(coeff)
+        self.play(ApplyMethod((self.axes).shift,3*LEFT),ApplyMethod(coeff.shift,3*LEFT),ApplyMethod(graph.shift,3*LEFT))      
+        topText=TextMobject("Here we get","output","for","any value of $|z|$").scale(0.4).shift(3*UP+3*RIGHT).set_color_by_tex_to_color_map({"output":YELLOW,"any value of $|z|$":BLUE})
+        topText1=TextMobject("Except for $|z|=0$").scale(0.7).shift(2.5*UP+3*RIGHT).set_color(RED)
+        dot1=Dot(color=WHITE,radius=0.06)
+        dot2=Dot(color=WHITE,radius=0.06)
+        self.play(Write(topText))
+        self.play(MoveAlongPath(dot1,graph),MoveAlongPath(dot2,xAxis),run_time=2)
+        self.play(Write(topText1))
+        self.play(FadeOut(dot1),FadeOut(dot2))
+        self.wait(0.5)
+        path=self.get_graph(lambda x:2+(3/x)+(4/x**2),x_min=1.24,x_max=0.8)
+        path1=self.get_graph(lambda x:0,x_min=1.24,x_max=0.8)
+        graphUpdated=self.get_graph(lambda x:2+(3/x)+(4/x**2),x_min=0.8,x_max=6,color=GREEN)
+        self.camera_frame.save_state()
+        self.play(FadeOut(graph),Write(graphUpdated))
+        self.play(self.camera_frame.set_width, 30,
+                    MoveAlongPath(dot1,path),MoveAlongPath(dot2,path1),run_time=2)
+        self.wait(1)
+
+        self.play(FadeOut(dot1),FadeOut(dot2),FadeOut(graphUpdated),FadeIn(graph),self.camera_frame.set_width,15)
+        self.wait(1)
+
+
+
+        
+
diff --git a/FSF-2020/calculus/series-and-transformations/Z-Transform/video3_RegionOfConvergence.py b/FSF-2020/calculus/series-and-transformations/Z-Transform/video3_RegionOfConvergence.py
new file mode 100644
index 0000000..bdfd8b3
--- /dev/null
+++ b/FSF-2020/calculus/series-and-transformations/Z-Transform/video3_RegionOfConvergence.py
@@ -0,0 +1,144 @@
+from manimlib.imports import *
+import numpy as np
+import math
+
+class graph1(GraphScene):
+    CONFIG = {
+        "x_min": -3,
+        "x_max": 5,
+        "y_min": -1,
+        "y_max": 1,
+        "graph_origin": ORIGIN,
+        "function_color": RED,
+        "axes_color": BLUE,
+        "x_axis_label": "$n$",
+        "y_axis_label": "$x(n)$",
+        "x_labeled_nums": range(-3, 6, 1),
+        "y_axis_height": 7,
+        "y_tick_frequency": 0.1,
+    }
+    def func(self,x,n):
+        summ=0
+        for i in range(n+1):
+            summ+=(1/(math.pow(x,i)))
+        return summ
+
+    def finalFunc(self,x):
+        if(x!=0):
+            return 1/(1-(1/(2*x)))
+            
+
+    def construct(self):
+        x_each_unit = self.x_axis_width / (self.x_max - self.x_min)
+        y_each_unit = self.y_axis_height / (self.y_max - self.y_min)   
+        self.setup_axes(animate=True,scalee=0.8)  
+        function=TextMobject("$X(t)=\sum _{ n=0 }^{ \infty  }{ { (0.5) }^{ n }{ z }^{ -n } }$").scale(0.4).shift(5*RIGHT+3*UP).set_color(RED)
+        self.play(FadeIn(function))
+        twoDGraph=[]
+        for i in range(5):
+            twoDGraph.append(Line(start=(i*x_each_unit,0,0),end=(i*x_each_unit,math.pow(0.5,i)*y_each_unit,0),color=GREEN))
+
+        groupGraph=VGroup(self.axes,twoDGraph[0],twoDGraph[1],twoDGraph[2],twoDGraph[3],twoDGraph[4])
+        self.play(Write(twoDGraph[0]),ShowCreation(twoDGraph[1]),ShowCreation(twoDGraph[2]),ShowCreation(twoDGraph[3]),ShowCreation(twoDGraph[4]))
+        self.wait(1.2)
+
+        self.play(ApplyMethod(groupGraph.scale,0.7))
+        self.play(ApplyMethod(groupGraph.shift,6*LEFT),ApplyMethod(function.move_to,5*LEFT+3*UP))
+        
+        someText1=TextMobject("Since it is a","summation","of","infinite terms",", it might").shift(2*RIGHT+2*UP).scale(0.5).set_color_by_tex_to_color_map({"summation":YELLOW,"infinite terms":BLUE})
+        someText2=TextMobject("Converge","or","Diverge").shift(2*RIGHT+0.5*DOWN+2*UP).scale(0.7).set_color_by_tex_to_color_map({"Converge":GREEN,"Diverge":RED})
+        someText3=TextMobject("depending upon","$|z|$").shift(2*RIGHT+UP).scale(0.5).set_color_by_tex_to_color_map({"$|z|$":YELLOW})
+        self.play(Write(someText1))
+        self.play(FadeIn(someText2))
+        self.play(Write(someText3))
+        self.wait(1)
+        self.play(FadeOut(someText1),FadeOut(someText2),FadeOut(someText3))
+
+        self.graph_origin=2*RIGHT+DOWN
+        self.x_axis_width=6
+        self.y_axis_height=5
+        self.y_tick_frequency=1
+        self.x_axis_label="$|z|$"
+        self.y_axis_label="$|X(n)|$"
+        self.x_min=-3
+        self.x_max=5
+        self.y_min=-1
+        self.y_max=5
+        self.x_labeled_nums=range(-3,6,1)
+        self.setup_axes(animate=True,scalee=0.6)
+        x_each_unit = self.x_axis_width / (self.x_max - self.x_min)
+        y_each_unit = self.y_axis_height / (self.y_max - self.y_min)   
+        rightSideGraphs=[]
+        xmins=[0,0.25,0.65,0.9,1]
+        for i in range(5):
+            rightSideGraphs.append(self.get_graph(lambda x:self.func(x,i),x_min=xmins[i],x_max=5,color=GREEN))
+        rightSideGraphs.append(self.get_graph(lambda x:1/(1-(1/(2*x))),x_min=0.63,x_max=5,color=GREEN))
+
+        graphCoeff=[
+                    TextMobject("$1$").scale(0.4).shift(self.graph_origin+x_each_unit*RIGHT*2+0.65*UP*y_each_unit*2+DOWN*y_each_unit*0.5).set_color(RED),
+                    TextMobject("$1+\\frac { 1 }{ 2|z| }$").scale(0.4).shift(self.graph_origin+x_each_unit*RIGHT*2+UP*y_each_unit).set_color(RED),
+                    TextMobject("$1+\\frac { 1 }{ 2|z| } +\\frac { 1 }{ { 2|z| }^{ 2 } } $").scale(0.4).shift(self.graph_origin+x_each_unit*RIGHT*2+UP*y_each_unit).set_color(RED),
+                    TextMobject("$1+\\frac { 1 }{ 2|z| } +\\frac { 1 }{ { (2|z|) }^{ 2 } } +\\frac { 1 }{ { (2|z|) }^{ 3 } }$").scale(0.4).shift(self.graph_origin+x_each_unit*RIGHT*2+UP*y_each_unit).set_color(RED),
+                    TextMobject("$1+\\frac { 1 }{ 2|z| } +\\frac { 1 }{ { (2|z|) }^{ 2 } } +\\frac { 1 }{ { (2|z|) }^{ 3 } } +\\frac { 1 }{ (2|z|)^{ 4 } } $").scale(0.4).shift(self.graph_origin+x_each_unit*RIGHT*2+UP*y_each_unit).set_color(RED),
+                    TextMobject("$\\frac { 1 }{ (1-\\frac { 1 }{ 2z } ) } $").scale(0.4).shift(self.graph_origin+x_each_unit*RIGHT*2+UP*y_each_unit).set_color(RED)
+        ]
+
+        self.play(ReplacementTransform(twoDGraph[0],rightSideGraphs[0]),FadeIn(graphCoeff[0]))
+        self.wait(0.5)
+        self.play(FadeOut(rightSideGraphs[0]),ReplacementTransform(twoDGraph[1],rightSideGraphs[1]),ReplacementTransform(graphCoeff[0],graphCoeff[1]))
+        self.wait(0.5)
+        self.play(FadeOut(rightSideGraphs[1]),ReplacementTransform(twoDGraph[2],rightSideGraphs[2]),ReplacementTransform(graphCoeff[1],graphCoeff[2]))
+        self.wait(0.5)
+        self.play(FadeOut(rightSideGraphs[2]),ReplacementTransform(twoDGraph[3],rightSideGraphs[3]),ReplacementTransform(graphCoeff[2],graphCoeff[3]))
+        self.wait(0.5)
+        self.play(FadeOut(rightSideGraphs[3]),ReplacementTransform(twoDGraph[4],rightSideGraphs[4]),ReplacementTransform(graphCoeff[3],graphCoeff[4]))
+        self.wait(0.5)
+        self.play(FadeOut(rightSideGraphs[4]),ShowCreation(rightSideGraphs[5]),ReplacementTransform(graphCoeff[4],graphCoeff[5]))
+        
+        self.wait(2)
+        # #self.play(FadeOut(self.axes),FadeOut(function),FadeOut(twoDGraph[0]),FadeOut(twoDGraph[1]),FadeOut(twoDGraph[2]))
+
+
+class graphCont(GraphScene,MovingCameraScene):
+    CONFIG = {
+        "x_min": -3,
+        "x_max": 5,
+        "y_min": -1,
+        "y_max": 5,
+        "graph_origin": 2*RIGHT+DOWN,
+        "function_color": RED,
+        "axes_color": BLUE,
+        "x_axis_label": "$|z|$",
+        "y_axis_label": "$|X(n)|$",
+        "x_labeled_nums": range(-3, 6, 1),
+        "x_axis_width": 6,
+        "y_axis_height": 5
+    }
+    def setup(self):
+        GraphScene.setup(self)
+        MovingCameraScene.setup(self)
+
+    def construct(self):
+        x_each_unit = self.x_axis_width / (self.x_max - self.x_min)
+        y_each_unit = self.y_axis_height / (self.y_max - self.y_min)   
+
+        coeff=TextMobject("$\\frac { 1 }{ (1-\\frac { 1 }{ 2z } ) } $").scale(0.4).shift(self.graph_origin+x_each_unit*RIGHT*2+UP*y_each_unit).set_color(RED)
+        self.setup_axes(scalee=0.6)
+        graph=self.get_graph(lambda x:1/(1-(1/(2*x))),x_min=0.63,x_max=5,color=GREEN)
+        
+        self.add(graph)
+        self.add(coeff)
+        
+        self.play(ApplyMethod((self.axes).shift,3*LEFT),ApplyMethod(coeff.shift,3*LEFT),ApplyMethod(graph.shift,3*LEFT))      
+        self.wait(1)
+
+        dashLine=DashedLine(start=self.graph_origin+3*LEFT+0.5*x_each_unit*RIGHT,end=self.graph_origin+3*LEFT+0.5*x_each_unit*RIGHT+y_each_unit*UP*5,color=YELLOW)
+        pt=TextMobject("0.5").scale(0.3).shift(self.graph_origin+3*LEFT+0.5*x_each_unit*RIGHT+DOWN*y_each_unit*0.3)
+        self.play(Write(dashLine))
+        self.play(Write(pt))
+        self.wait(0.6)
+        rectRegion=Rectangle(height=y_each_unit*5,width=x_each_unit*5,fill_color=WHITE,fill_opacity=0.3,opacity=0.3,color=BLACK).shift(1.6*RIGHT*x_each_unit+0.5*DOWN*y_each_unit+1.5*UP)
+        self.play(ShowCreation(rectRegion))
+        text=TextMobject("Region Of Convergence!").scale(0.4).shift(4.6*RIGHT+1.5*UP).set_color(GREEN)
+        self.play(FadeIn(text))
+        self.wait(2)