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Laplace Transform
When we talk about the Laplace transform, we are actually talking about the version of the Laplace transform known as the unilinear Laplace Transform. The other version, the Bilinear Laplace Transform (not related to the Bilinear Transform, below) is not used in this book.
The Laplace Transform is defined as:
[Laplace Transform]
F ( s ) = ℒ [ f ( t ) ] = {∫}_{− ∞}^{∞} x ( t ) {e}^{− s t} d t
And the Inverse Laplace Transform is defined as:
[Inverse Laplace Transform]
f ( t ) = {ℒ}^{− 1} \{ F ( s ) \} = \frac{1}{2 π j} {∫}_{σ − j ∞}^{σ + j ∞} X ( s ) {e}^{s t} d s
Table of Laplace Transforms
This is a table of common Laplace Transforms.
| No. | Time Domain x ( t ) = {ℒ}^{− 1} \{ X ( s ) \} | Laplace Domain X ( s ) = ℒ \{ x ( t ) \} |
|---|---|---|
| 1 | \frac{1}{2 π j} {∫}_{σ − j ∞}^{σ + j ∞} X ( s ) {e}^{s t} d s < m a t h > | < m a t h > {∫}_{− ∞}^{∞} x ( t ) {e}^{− s t} d t | |
| 2 | $δ ( t ) $ | ${}1 $ |
| 3 | $δ ( t − a ) $ | ${e}^{− a s} $ |
| 4 | $u ( t ) $ | \frac{1}{s} |
| 5 | $u ( t − a ) $ | \frac{{e}^{− a s}}{s} |
| 6 | $t u ( t ) $ | \frac{1}{{s}^{2}} |
| 7 | ${t}^{n} u ( t ) $ | \frac{n !}{{s}^{n + 1}} |
| 8 | \frac{1}{\sqrt{π t}} u ( t ) | \frac{1}{\sqrt{s}} |
| 9 | ${e}^{a t} u ( t ) $ | \frac{1}{s − a} |
| 10 | ${t}^{n} {e}^{a t} u ( t ) $ | \frac{n !}{( s − a {)}^{n + 1}} |
| 11 | $cos ( ω t ) u ( t ) $ | \frac{s}{{s}^{2} + {ω}^{2}} |
| 12 | $sin ( ω t ) u ( t ) $ | \frac{ω}{{s}^{2} + {ω}^{2}} |
| 13 | $cosh ( ω t ) u ( t ) $ | \frac{s}{{s}^{2} − {ω}^{2}} |
| 14 | $sinh ( ω t ) u ( t ) $ | \frac{ω}{{s}^{2} − {ω}^{2}} |
| 15 | ${e}^{a t} cos ( ω t ) u ( t ) $ | \frac{s − a}{( s − a {)}^{2} + {ω}^{2}} |
| 16 | ${e}^{a t} sin ( ω t ) u ( t ) $ | \frac{ω}{( s − a {)}^{2} + {ω}^{2}} |
| 17 | \frac{1}{2 {ω}^{3}} ( sin ω t − ω t cos ω t ) | \frac{1}{( {s}^{2} + {ω}^{2} {)}^{2}} |
| 18 | \frac{t}{2 ω} sin ω t | \frac{s}{( {s}^{2} + {ω}^{2} {)}^{2}} |
| 19 | \frac{1}{2 ω} ( sin ω t + ω t cos ω t ) | \frac{{s}^{2}}{( {s}^{2} + {ω}^{2} {)}^{2}} |
Properties of the Laplace Transform
This is a table of the most important properties of the laplace transform.
| Property | Definition |
|---|---|
| Linearity | ℒ \{ a f ( t ) + b g ( t ) \} = a F ( s ) + b G ( s ) |
ℒ \{ {f}^{″} \} = {s}^{2} ℒ \{ f \} − s f ( {0}^{−} ) − {f}^{′} ( {0}^{−} )
ℒ \{ {f}^{( n )} \} = {s}^{n} ℒ \{ f \} − {s}^{n − 1} f ( {0}^{−} ) − ⋯ − {f}^{( n − 1 )} ( {0}^{−} ) |
| Frequency Division | ℒ \{ t f ( t ) \} = − {F}^{′} ( s )ℒ \{ {t}^{n} f ( t ) \} = ( − 1 {)}^{n} {F}^{( n )} ( s ) |
| Frequency Integration | ℒ \{ \frac{f ( t )}{t} \} = {∫}_{s}^{∞} F ( σ ) \ d σ |
| Time Integration | ℒ \{ {∫}_{0}^{t} f ( τ ) \ d τ \} = ℒ \{ u ( t ) ∗ f ( t ) \} = \frac{1}{s} F ( s ) |
| Scaling | ℒ \{ f ( a t ) \} = \frac{1}{a} F ( \frac{s}{a} ) |
| Initial value theorem | f ( {0}^{+} ) = {lim}_{s → ∞} s F ( s ) |
| Final value theorem | f ( ∞ ) = {lim}_{s → 0} s F ( s ) |
| Frequency Shifts | ℒ \{ {e}^{a t} f ( t ) \} = F ( s − a ){ℒ}^{− 1} \{ F ( s − a ) \} = {e}^{a t} f ( t ) |
| Time Shifts | ℒ \{ f ( t − a ) u ( t − a ) \} = {e}^{− a s} F ( s ){ℒ}^{− 1} \{ {e}^{− a s} F ( s ) \} = f ( t − a ) u ( t − a ) |
| Convolution Theorem | ℒ \{ f ( t ) ∗ g ( t ) \} = F ( s ) G ( s ) |
Where:
f ( t ) = {ℒ}^{− 1} \{ F ( s ) \}
g ( t ) = {ℒ}^{− 1} \{ G ( s ) \}
s = σ + j ω
Fourier Transform
The Fourier Transform is used to break a time-domain signal into its frequency domain components. The Fourier Transform is very closely related to the Laplace Transform, and is only used in place of the Laplace transform when the system is being analyzed in a frequency context.
The Fourier Transform is defined as:
[Fourier Transform]
F ( j ω ) = ℱ [ f ( t ) ] = {∫}_{0}^{∞} f ( t ) {e}^{− j ω t} d t
And the Inverse Fourier Transform is defined as:
[Inverse Fourier Transform]
f ( t ) = {ℱ}^{− 1} \{ F ( j ω ) \} = \frac{1}{2 π} {∫}_{− ∞}^{∞} F ( j ω ) {e}^{− j ω t} d ω
Table of Fourier Transforms
This is a table of common fourier transforms.
| Time Domain | Frequency Domain | |
|---|---|---|
x ( t ) = {ℱ}^{− 1} \{ X ( ω ) \} | X ( ω ) = ℱ \{ x ( t ) \} | |
| 1 | X ( j ω ) = {∫}_{− ∞}^{∞} x ( t ) {e}^{− j ω t} d t | x ( t ) = \frac{1}{2 π} {∫}_{− ∞}^{∞} X ( ω ) {e}^{j ω t} d ω |
| 2 | ${}1 $ | ${}2 π δ ( ω ) $ |
| 3 | $− 0.5 + u ( t ) $ | $\frac{1}{j ω} $ |
| 4 | $δ ( t ) $ | ${}1 $ |
| 5 | $δ ( t − c ) $ | ${e}^{− j ω c} $ |
| 6 | $u ( t ) $ | $π δ ( ω ) + \frac{1}{j ω} $ |
| 7 | {e}^{− b t} u ( t ) \ ( b > 0 ) | $\frac{1}{j ω + b} $ |
| 8 | $cos {ω}_{0} t $ | $π [ δ ( ω + {ω}{0} ) + δ ( ω − {ω}{0} ) ] $ |
| 9 | $cos ( {ω}_{0} t + θ ) $ | $π [ {e}^{− j θ} δ ( ω + {ω}{0} ) + {e}^{j θ} δ ( ω − {ω}{0} ) ] $ |
| 10 | $sin {ω}_{0} t $ | $j π [ δ ( ω + {ω}{0} ) − δ ( ω − {ω}{0} ) ] $ |
| 11 | $sin ( {ω}_{0} t + θ ) $ | $j π [ {e}^{− j θ} δ ( ω + {ω}{0} ) − {e}^{j θ} δ ( ω − {ω}{0} ) ] $ |
| 12 | $\text{rect} ( \frac{t}{τ} ) $ | $τ \text{sinc} ( \frac{τ ω}{2 π} ) $ |
| 13 | $τ \text{sinc} ( \frac{τ t}{2 π} ) $ | ${}2 π \text{rect} ( \frac{ω}{τ} ) $ |
| 14 | $( 1 − \frac{2 | t |}{τ} ) \text{rect} ( \frac{t}{τ} ) $ | $\frac{τ}{2} {\text{sinc}}^{2} ( \frac{τ ω}{4 π} ) $ |
| 15 | $\frac{τ}{2} {\text{sinc}}^{2} ( \frac{τ t}{4 π} ) $ | ${}2 π ( 1 − \frac{2 | ω |}{τ} ) \text{rect} ( \frac{ω}{τ} ) $ |
| 16 | ${e}^{− a | t |} , ℜ { a } > 0 $ | $\frac{2 a}{{a}^{2} + {ω}^{2}} $ |
\text{rect} ( \frac{t}{τ} ) is the rectangular pulse function of width τu ( t ) is the Heaviside step functionδ ( t ) is the Dirac delta function | |
| This box: view talk edit | | |
Table of Fourier Transform Properties
This is a table of common properties of the fourier transform.
| Signal | Fourier transform unitary, angular frequency | Fourier transform unitary, ordinary frequency | Remarks | |
|---|---|---|---|---|
| $g ( t ) \ ≡ $ $\frac{1}{\sqrt{2 π}} {∫}_{− ∞}^{∞} \ \ G ( ω ) {e}^{i ω t} d ω $ | $G ( ω ) \ ≡ $ $\frac{1}{\sqrt{2 π}} {∫}_{− ∞}^{∞} \ \ g ( t ) {e}^{− i ω t} d t $ | G ( f ) \ ≡ ${∫}_{− ∞}^{∞} \ \ g ( t ) {e}^{− i 2 π f t} d t $ | ||
| 1 | $a ⋅ g ( t ) + b ⋅ h ( t ) $ | $a ⋅ G ( ω ) + b ⋅ H ( ω ) $ | $a ⋅ G ( f ) + b ⋅ H ( f ) $ | Linearity |
| 2 | $g ( t − a ) $ | ${e}^{− i a ω} G ( ω ) $ | ${e}^{− i 2 π a f} G ( f ) $ | Shift in time domain |
| 3 | ${e}^{i a t} g ( t ) $ | $G ( ω − a ) $ | $G ( f − \frac{a}{2 π} ) $ | Shift in frequency domain, dual of 2 |
| 4 | $g ( a t ) $ | $\frac{1}{| a |} G ( \frac{ω}{a} ) $ | $\frac{1}{| a |} G ( \frac{f}{a} ) $ | If $| a | $ is large, then $g ( a t ) $ is concentrated around 0 and $\frac{1}{| a |} G ( \frac{ω}{a} ) $ spreads out and flattens |
| 5 | $G ( t ) $ | $g ( − ω ) $ | $g ( − f ) $ | Duality property of the Fourier transform. Results from swapping "dummy" variables of $t $ and $ω $. |
| 6 | $\frac{{d}^{n} g ( t )}{d {t}^{n}} $ | $( i ω {)}^{n} G ( ω ) $ | $( i 2 π f {)}^{n} G ( f ) $ | Generalized derivative property of the Fourier transform |
| 7 | ${t}^{n} g ( t ) $ | ${i}^{n} \frac{{d}^{n} G ( ω )}{d {ω}^{n}} $ | ${( \frac{i}{2 π} )}^{n} \frac{{d}^{n} G ( f )}{d {f}^{n}} $ | This is the dual to 6 |
| 8 | $( g ∗ h ) ( t ) $ | $\sqrt{2 π} G ( ω ) H ( ω ) $ | $G ( f ) H ( f ) $ | $g ∗ h $ denotes the convolution of $g $ and $h $ — this rule is the convolution theorem |
| 9 | $g ( t ) h ( t ) $ | \frac{( G ∗ H ) ( ω )}{\sqrt{2 π} \ } | $( G ∗ H ) ( f ) $ | This is the dual of 8 |
| 10 | For a purely real even function $g ( t ) $ | $G ( ω ) $ is a purely real even function | $G ( f ) $ is a purely real even function | |
| 11 | For a purely real odd function $g ( t ) $ | $G ( ω ) $ is a purely imaginary odd function | $G ( f ) $ is a purely imaginary odd function |
Z-Transform
The Z-transform is used primarily to convert discrete data sets into a continuous representation. The Z-transform is notationally very similar to the star transform, except that the Z transform does not take explicit account for the sampling period. The Z transform has a number of uses in the field of digital signal processing, and the study of discrete signals in general, and is useful because Z-transform results are extensively tabulated, whereas star-transform results are not.
The Z Transform is defined as:
[Z Transform]
X ( z ) = 𝒵 [ x [ n ] ] = {∑}_{n = − ∞}^{∞} x [ n ] {z}^{− n}
Inverse Z Transform
The inverse Z Transform is a highly complex transformation, and might be inaccessible to students without enough background in calculus. However, students who are familiar with such integrals are encouraged to perform some inverse Z transform calculations, to verify that the formula produces the tabulated results.
[Inverse Z Transform]
x [ n ] = \frac{1}{2 π j} {∮}_{C} X ( z ) {z}^{n − 1} d z
Z-Transform Tables
Here:
u [ n ] = 1 for n > = 0, u [ n ] = 0 for n < 0
δ [ n ] = 1forn = 0,δ [ n ] = 0otherwise
Signal, x [ n ] | Z-transform, X ( z ) | ROC | |
|---|---|---|---|
| 1 | $δ [ n ] $ | ${}1 $ | $\text{all } z $ |
| 2 | $δ [ n − {n}_{0} ] $ | ${z}^{− {n}_{0}} $ | $z ≠ 0 $ |
| 3 | $u [ n ] $ | \frac{1}{1 − {z}^{− 1}} | $| z | > 1 $ |
| 4 | $− u [ − n − 1 ] $ | \frac{1}{1 − {z}^{− 1}} | $| z | < 1 $ |
| 5 | $n u [ n ] $ | \frac{{z}^{− 1}}{( 1 − {z}^{− 1} {)}^{2}} | $| z | > 1 $ |
| 6 | $− n u [ − n − 1 ] $ | \frac{{z}^{− 1}}{( 1 − {z}^{− 1} {)}^{2}} | $| z | < 1 $ |
| 7 | ${n}^{2} u [ n ] $ | \frac{{z}^{− 1} ( 1 + {z}^{− 1} )}{( 1 − {z}^{− 1} {)}^{3}} | $| z | > 1 $ |
| 8 | $− {n}^{2} u [ − n − 1 ] $ | \frac{{z}^{− 1} ( 1 + {z}^{− 1} )}{( 1 − {z}^{− 1} {)}^{3}} | $| z | < 1 $ |
| 9 | ${n}^{3} u [ n ] $ | \frac{{z}^{− 1} ( 1 + 4 {z}^{− 1} + {z}^{− 2} )}{( 1 − {z}^{− 1} {)}^{4}} | $| z | > 1 $ |
| 10 | $− {n}^{3} u [ − n − 1 ] $ | \frac{{z}^{− 1} ( 1 + 4 {z}^{− 1} + {z}^{− 2} )}{( 1 − {z}^{− 1} {)}^{4}} | $| z | < 1 $ |
| 11 | ${a}^{n} u [ n ] $ | \frac{1}{1 − a {z}^{− 1}} | $| z | > | a | $ |
| 12 | $− {a}^{n} u [ − n − 1 ] $ | \frac{1}{1 − a {z}^{− 1}} | $| z | < | a | $ |
| 13 | $n {a}^{n} u [ n ] $ | \frac{a {z}^{− 1}}{( 1 − a {z}^{− 1} {)}^{2}} | $| z | > | a | $ |
| 14 | $− n {a}^{n} u [ − n − 1 ] $ | \frac{a {z}^{− 1}}{( 1 − a {z}^{− 1} {)}^{2}} | $| z | < | a | $ |
| 15 | ${n}^{2} {a}^{n} u [ n ] $ | \frac{a {z}^{− 1} ( 1 + a {z}^{− 1} )}{( 1 − a {z}^{− 1} {)}^{3}} | $| z | > | a | $ |
| 16 | $− {n}^{2} {a}^{n} u [ − n − 1 ] $ | \frac{a {z}^{− 1} ( 1 + a {z}^{− 1} )}{( 1 − a {z}^{− 1} {)}^{3}} | $| z | < | a | $ |
| 17 | $cos ( {ω}_{0} n ) u [ n ] $ | \frac{1 − {z}^{− 1} cos ( {ω}_{0} )}{1 − 2 {z}^{− 1} cos ( {ω}_{0} ) + {z}^{− 2}} | $| z | > 1 $ |
| 18 | $sin ( {ω}_{0} n ) u [ n ] $ | \frac{{z}^{− 1} sin ( {ω}_{0} )}{1 − 2 {z}^{− 1} cos ( {ω}_{0} ) + {z}^{− 2}} | $| z | > 1 $ |
| 19 | ${a}^{n} cos ( {ω}_{0} n ) u [ n ] $ | \frac{1 − a {z}^{− 1} cos ( {ω}_{0} )}{1 − 2 a {z}^{− 1} cos ( {ω}_{0} ) + {a}^{2} {z}^{− 2}} | $| z | > | a | $ |
| 20 | ${a}^{n} sin ( {ω}_{0} n ) u [ n ] $ | \frac{a {z}^{− 1} sin ( {ω}_{0} )}{1 − 2 a {z}^{− 1} cos ( {ω}_{0} ) + {a}^{2} {z}^{− 2}} | $| z | > | a | $ |
Modified Z-Transform
The Modified Z-Transform is similar to the Z-transform, except that the modified version allows for the system to be subjected to any arbitrary delay, by design. The Modified Z-Transform is very useful when talking about digital systems for which the processing time of the system is not negligible. For instance, a slow computer system can be modeled as being an instantaneous system with an output delay.
The modified Z transform is based off the delayed Z transform:
[Modified Z Transform]
X ( z , m ) = X ( z , Δ ) {|}_{Δ → 1 − m} = 𝒵 \{ X ( s ) {e}^{− Δ T s} \} {|}_{Δ → 1 − m}
Star Transform
The Star Transform is a discrete transform that has similarities between the Z transform and the Laplace Transform. In fact, the Star Transform can be said to be nearly analogous to the Z transform, except that the Star transform explicitly accounts for the sampling time of the sampler.
The Star Transform is defined as:
[Star Transform]
{F}^{∗} ( s ) = {ℒ}^{∗} [ f ( t ) ] = {∑}_{k = 0}^{∞} f ( k T ) {e}^{− s k T}
Star transform pairs can be obtained by plugging z = {e}^{s T} into the Z-transform pairs, above.
Bilinear Transform
The bilinear transform is used to convert an equation in the Z domain into the arbitrary W domain, with the following properties:
roots inside the unit circle in the Z-domain will be mapped to roots on the left-half of the W plane.
- roots outside the unit circle in the Z-domain will be mapped to roots on the right-half of the W plane
- roots on the unit circle in the Z-domain will be mapped onto the vertical axis in the W domain.
The bilinear transform can therefore be used to convert a Z-domain equation into a form that can be analyzed using the Routh-Hurwitz criteria. However, it is important to note that the W-domain is not the same as the complex Laplace S-domain. To make the output of the bilinear transform equal to the S-domain, the signal must be prewarped, to account for the non-linear nature of the bilinear transform.
The Bilinear transform can also be used to convert an S-domain system into the Z domain. Again, the input system must be prewarped prior to applying the bilinear transform, or else the results will not be correct.
The Bilinear transform is governed by the following variable transformations:
[Bilinear Transform]
z = \frac{( T / 2 ) + w}{( T / 2 ) − w} , \ w = \frac{2}{T} \frac{z − 1}{z + 1}
Where T is the sampling time of the discrete signal.
Frequencies in the w domain are related to frequencies in the s domain through the following relationship:
{ω}_{w} = \frac{2}{T} tan ( \frac{{ω}_{s} T}{2} )
This relationship is called the frequency warping characteristic of the bilinear transform. To counter-act the effects of frequency warping, we can pre-warp the Z-domain equation using the inverse warping characteristic. If the equation is prewarped before it is transformed, the resulting poles of the system will line up more faithfully with those in the s-domain.
[Bilinear Frequency Prewarping]
ω = \frac{2}{T} arctan ( {ω}_{a} \frac{T}{2} ) .
Applying these transformations before applying the bilinear transform actually enables direct conversions between the S-Domain and the Z-Domain. The act of applying one of these frequency warping characteristics to a function before transforming is called prewarping.
Wikipedia Resources
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