Fourier Transforms/Singularity
General Fourier Transform of Singularity Function Combinations
Consider a continuous-time signal constructed from a linear combination of $ K $ scaled and time-shifted causal singularity functions (repeated integrals of the unit impulse):
- $ \displaystyle x(t) = \sum_{k=1}^{K} A_k \, u_{n_k}(t - t_k) = \sum_{k=1}^{K} A_k \frac{(t - t_k)^{n_k - 1}}{(n_k - 1)!} u(t - t_k) $
where:
- $ A_k \in \mathbb{R} $ is the amplitude scaling coefficient of the $ k $-th component
- $ t_k \in \mathbb{R} $ is the time-shift (activation time) of the $ k $-th component
- $ n_k \in \{1, 2, 3, \dots\} $ denotes the singularity order:
- $ u_1(t) = u(t) $ is the unit step function
- $ u_2(t) = r(t) = t\,u(t) $ is the unit ramp function
- $ u_3(t) = \frac{1}{2}t^2 u(t) $ is the causal parabolic function
- In general, $ u_n(t) = \frac{t^{n-1}}{(n-1)!}u(t) $ for $ n \ge 1 $
Master Transform Formula
The continuous-time Fourier transform decomposes into a strictly dynamic rational component and a singular boundary component involving Dirac deltas and their distributional derivatives:
- $ X(j\omega) = X_{\text{rational}}(j\omega) + X_{\text{singular}}(\omega) $
- $ \displaystyle X(j\omega) = \sum_{k=1}^{K} \frac{A_k \, e^{-j\omega t_k}}{(j\omega)^{n_k}} + \pi \sum_{m=0}^{N_{\max}-1} \frac{j^m}{m!} C_m \, \delta^{(m)}(\omega) $
where:
- $ N_{\max} = \max_k(n_k) $ is the highest singularity order present among all terms
- $ \delta^{(m)}(\omega) = \frac{d^m}{d\omega^m}\delta(\omega) $ denotes the $ m $-th distributional derivative of the Dirac delta
- The scalar coefficient $ C_m $ weighting each derivative order is:
- $ \displaystyle C_m = \sum_{n=m+1}^{N_{\max}} \frac{1}{(n - m - 1)!} \sum_{k=1}^{K} A_k (-t_k)^{n - m - 1} \delta[n_k - n] $
with $ \delta[n_k - n] $ representing the discrete Kronecker delta that selects only components of singularity order $ n $.
Unpacked Coefficients by Order
For practical problems involving combinations of steps ($ n=1 $), ramps ($ n=2 $), and parabolas ($ n=3 $):
| Singularity Index | Delta Derivative | Singular Contribution | Coefficient Formula $ C_m $ |
|---|---|---|---|
| $ m = 0 $ | $ \delta(\omega) $ | $ \pi C_0 \, \delta(\omega) $ | $ \displaystyle C_0 = \left.\sum_{k=1}^K A_k\right|_{n_k=1} + \left.\sum_{k=1}^K A_k(-t_k)\right|_{n_k=2} + \left.\sum_{k=1}^K A_k \frac{t_k^2}{2}\right|_{n_k=3} + \dots $ |
| $ m = 1 $ | $ \delta'(\omega) $ | $ j\pi C_1 \, \delta'(\omega) $ | $ \displaystyle C_1 = \left.\sum_{k=1}^K A_k\right|_{n_k=2} + \left.\sum_{k=1}^K A_k(-t_k)\right|_{n_k=3} + \dots $ |
| $ m = 2 $ | $ \delta''(\omega) $ | $ -\dfrac{\pi}{2} C_2 \, \delta''(\omega) $ | $ \displaystyle C_2 = \left.\sum_{k=1}^K A_k\right|_{n_k=3} + \dots $ |
Worked Examples
Example 1: Steps Only, Zero to Zero
Consider a pulse composed solely of steps that starts at zero and returns to zero:
- $ \displaystyle x_1(t) = 2u(t+1) - 5u(t-1) + 3u(t-4) $
Evaluating the parameters:
- Number of terms: $ K = 3 $
- Singularity orders: $ n_1 = 1, n_2 = 1, n_3 = 1 \implies N_{\max} = 1 $
- Amplitudes: $ A_1 = 2, A_2 = -5, A_3 = 3 $
- Shifts: $ t_1 = -1, t_2 = 1, t_3 = 4 $
1. Singular Term ($ m = 0 $):
Since $ N_{\max} = 1 $, the only possible delta term is $ m = 0 $ ($ \delta(\omega) $):
- $ \displaystyle C_0 = \left.\sum_{k=1}^3 A_k\right|_{n_k=1} = 2 + (-5) + 3 = 0 $
The impulse vanishes completely: $ \pi C_0 \delta(\omega) = 0 $.
2. Rational Component:
- $ \displaystyle X_{\text{rational}}(j\omega) = \sum_{k=1}^3 \frac{A_k e^{-j\omega t_k}}{j\omega} = \frac{2e^{j\omega} - 5e^{-j\omega} + 3e^{-j4\omega}}{j\omega} $
3. Total Transform:
- $ \displaystyle X_1(j\omega) = \frac{2e^{j\omega} - 5e^{-j\omega} + 3e^{-j4\omega}}{j\omega} $
- (Note: by L'Hôpital's Rule, $ \lim_{\omega \to 0} X_1(j\omega) = \int_{-\infty}^{\infty} x_1(t)\,dt = 2(2) + (-3)(3) = -5 $, confirming the singularity at $ \omega = 0 $ is removable).*
Example 2: Steps Only, Enters at 3 and Leaves at 1
Consider a signal that has a pre-existing DC offset of 3 as $ t \to -\infty $ and transitions to 1 as $ t \to +\infty $:
- $ \displaystyle x_2(t) = 3 + 2u(t+2) - 4u(t-3) $
Evaluating the terms:
- Constant DC term: $ \mathcal{F}\{3\} = 6\pi\delta(\omega) $
- Singularity sum: $ \tilde{x}_2(t) = 2u(t+2) - 4u(t-3) $
- $ n_1 = 1, A_1 = 2, t_1 = -2 $
- $ n_2 = 1, A_2 = -4, t_2 = 3 $
- $ N_{\max} = 1 $
1. Singular Term for the Steps ($ m = 0 $):
- $ \displaystyle C_0 = \left.\sum_{k=1}^2 A_k\right|_{n_k=1} = 2 + (-4) = -2 $
This contributes $ \pi C_0 \delta(\omega) = -2\pi\delta(\omega) $.
2. Combining with the Background DC:
The total singular coefficient is:
- $ \displaystyle X_{\text{singular}}(\omega) = 6\pi\delta(\omega) - 2\pi\delta(\omega) = 4\pi\delta(\omega) $
- (Check: the asymptotic average is $ \frac{x(-\infty) + x(+\infty)}{2} = \frac{3 + 1}{2} = 2 \implies 2\pi(2)\delta(\omega) = 4\pi\delta(\omega) $).*
3. Rational Component and Total Transform:
- $ \displaystyle X_2(j\omega) = 4\pi\delta(\omega) + \frac{2e^{j2\omega} - 4e^{-j3\omega}}{j\omega} $
Example 3: Steps and Ramps, Flat-to-Flat (Different Levels)
Consider a signal starting at 0 for $ t < -1 $, rising with a ramp, leveling off, and ending flat at a non-zero level of 2 for $ t \ge 3 $:
- $ \displaystyle x_3(t) = r(t+1) - r(t-1) - u(t-1) + u(t-3) $
Evaluating the parameters ($ K = 4 $):
| $ k $ | Term | $ n_k $ | $ A_k $ | $ t_k $ |
|---|---|---|---|---|
| 1 | $ r(t+1) $ | 2 | 1 | -1 |
| 2 | $ -r(t-1) $ | 2 | -1 | 1 |
| 3 | $ -u(t-1) $ | 1 | -1 | 1 |
| 4 | $ u(t-3) $ | 1 | 1 | 3 |
Here $ N_{\max} = 2 $, so we evaluate $ m = 1 $ ($ \delta' $) and $ m = 0 $ ($ \delta $).
1. Doublet Term ($ m = 1 $):
- $ \displaystyle C_1 = \left.\sum_{k=1}^4 A_k\right|_{n_k=2} = 1 + (-1) = 0 $
Because the incoming and outgoing slopes are both 0, $ C_1 = 0 $, and the doublet vanishes: $ j\pi C_1 \delta'(\omega) = 0 $.
2. Impulse Term ($ m = 0 $):
- $ \displaystyle C_0 = \left.\sum_{k=1}^4 A_k\right|_{n_k=1} + \left.\sum_{k=1}^4 A_k(-t_k)\right|_{n_k=2} $
- Step contribution ($ n_k = 1 $): $ (-1) + 1 = 0 $
- Ramp contribution ($ n_k = 2 $): $ (1)(-(-1)) + (-1)(-(1)) = 1(1) + (-1)(-1) = 1 + 1 = 2 $
Thus:
- $ \displaystyle C_0 = 0 + 2 = 2 \implies \pi C_0 \delta(\omega) = 2\pi\delta(\omega) $
- (Check: $ x(-\infty) = 0, x(+\infty) = 2 \implies x_{\text{DC}} = \frac{0 + 2}{2} = 1 \implies 2\pi(1)\delta(\omega) = 2\pi\delta(\omega) $).*
3. Rational Component and Total Transform:
- $ \displaystyle X_3(j\omega) = 2\pi\delta(\omega) + \frac{-e^{-j\omega} + e^{-j3\omega}}{j\omega} + \frac{e^{j\omega} - e^{-j\omega}}{(j\omega)^2} $
Example 4: Steps and Ramps, Flat to Constant Slope of 2
Consider a signal starting at 0 for $ t < 0 $, stepping up, and then continuing with an asymptotic slope of 2 for $ t \ge 2 $:
- $ \displaystyle x_4(t) = 3u(t) + 2r(t-2) $
Evaluating the parameters ($ K = 2 $):
- $ k=1 $: $ 3u(t) \implies n_1 = 1, A_1 = 3, t_1 = 0 $
- $ k=2 $: $ 2r(t-2) \implies n_2 = 2, A_2 = 2, t_2 = 2 $
- $ N_{\max} = 2 $
1. Doublet Term ($ m = 1 $):
- $ \displaystyle C_1 = \left.\sum_{k=1}^2 A_k\right|_{n_k=2} = 2 $
This yields a non-zero doublet contribution:
- $ \displaystyle j\pi C_1 \delta'(\omega) = 2j\pi \delta'(\omega) $
2. Impulse Term ($ m = 0 $):
- $ \displaystyle C_0 = \left.\sum_{k=1}^2 A_k\right|_{n_k=1} + \left.\sum_{k=1}^2 A_k(-t_k)\right|_{n_k=2} = 3 + 2(-2) = 3 - 4 = -1 $
This yields an impulse contribution:
- $ \displaystyle \pi C_0 \delta(\omega) = -\pi\delta(\omega) $
3. Rational Component:
- $ \displaystyle X_{\text{rational}}(j\omega) = \frac{3}{j\omega} + \frac{2e^{-j2\omega}}{(j\omega)^2} $
4. Total Transform:
- $ \displaystyle X_4(j\omega) = 2j\pi\delta'(\omega) - \pi\delta(\omega) + \frac{3}{j\omega} + \frac{2e^{-j2\omega}}{(j\omega)^2} $
References and Further Reading
- Oppenheim, A. V., Willsky, A. S., & Nawab, S. H. Signals and Systems, 2nd ed., Prentice Hall, 1996. (Chapter 4: The Continuous-Time Fourier Transform — covers the transform of the unit step $ u(t) $, frequency differentiation, and time-shift properties).
- Bracewell, R. N. The Fourier Transform and Its Applications, 3rd ed., McGraw-Hill, 1999. (Covers generalized functions, the doublet $ \delta' $, and derivative operations on step and ramp discontinuities).
- Strichartz, R. S. A Guide to Distribution Theory and Fourier Transforms, World Scientific, 2003. (Rigorous distributional framework for transforms of polynomials multiplied by the Heaviside step, $ \mathcal{S}' $ tempered distributions, and delta derivatives).
- Hoskins, R. F. Generalized Functions, Ellis Horwood / John Wiley & Sons, 1979. (Covers algebraic properties of distributions with point supports, including Taylor expansions of $ f(\omega)\delta^{(m)}(\omega) $).