3.3 Fourier Series, Fourier Transforms & Z-Transforms

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3.3 Fourier Series, Fourier Transforms & Z-Transforms

Key Takeaways Explain This Topic AI

Trigonometric and exponential Fourier series decompose arbitrary periodic waveforms into infinite sums of harmonically related sinusoidal components.

Waveform symmetries (even, odd, and half-wave symmetry) systematically eliminate specific harmonic terms, simplifying Fourier series coefficient evaluation.

The Continuous-Time Fourier Transform (CTFT) maps non-periodic continuous signals into the continuous frequency domain, with convolution in time mapping to multiplication in frequency.

The Z-transform provides the mathematical foundation for discrete-time signal processing, mapping difference equations into z-domain transfer functions (H(z)).

System stability for causal discrete-time systems requires all poles of (H(z)) to lie strictly inside the unit circle in the z-plane ((|z| < 1)).

Last updated: July 2026

3.3 Fourier Series, Fourier Transforms & Z-Transforms

Spectral frequency analysis allows communications and signal processing engineers to evaluate how signals distribute energy across the frequency spectrum. This section covers continuous-time periodic representations using Fourier Series, non-periodic continuous-time signals via the Continuous-Time Fourier Transform (CTFT), and discrete-time signals using the Z-Transform.

1. Fourier Series Representation of Periodic Signals

Any physically realizable periodic signal f ( t ) f(t) f ( t ) with period T T T and fundamental frequency ω 0 = 2 π T \omega_0 = \frac{2\pi}{T} ω 0 ​ = T 2 π ​ satisfying the Dirichlet conditions can be expanded into an infinite series of sinusoids.

1.1 Trigonometric Fourier Series Form

The Trigonometric Fourier Series expansion is defined as: f ( t ) = a 0 + ∑ n = 1 ∞ ( a n cos ⁡ ( n ω 0 t ) + b n sin ⁡ ( n ω 0 t ) ) f(t) = a_0 + \sum_{n=1}^{\infty} \left( a_n \cos(n\omega_0 t) + b_n \sin(n\omega_0 t) \right) f ( t ) = a 0 ​ + n = 1 ∑ ∞ ​ ( a n ​ cos ( n ω 0 ​ t ) + b n ​ sin ( n ω 0 ​ t ) )

Where the Fourier coefficients are evaluated via the integrals: a 0 = 1 T ∫ 0 T f ( t ) d t ( DC Average Value ) a_0 = \frac{1}{T} \int_{0}^{T} f(t) dt \quad (\text{DC Average Value}) a 0 ​ = T 1 ​ ∫ 0 T ​ f ( t ) d t ( DC Average Value ) a n = 2 T ∫ 0 T f ( t ) cos ⁡ ( n ω 0 t ) d t ( n = 1 , 2 , 3 , … ) a_n = \frac{2}{T} \int_{0}^{T} f(t) \cos(n\omega_0 t) dt \quad (n = 1, 2, 3, \dots) a n ​ = T 2 ​ ∫ 0 T ​ f ( t ) cos ( n ω 0 ​ t ) d t ( n = 1 , 2 , 3 , … ) b n = 2 T ∫ 0 T f ( t ) sin ⁡ ( n ω 0 t ) d t ( n = 1 , 2 , 3 , … ) b_n = \frac{2}{T} \int_{0}^{T} f(t) \sin(n\omega_0 t) dt \quad (n = 1, 2, 3, \dots) b n ​ = T 2 ​ ∫ 0 T ​ f ( t ) sin ( n ω 0 ​ t ) d t ( n = 1 , 2 , 3 , … )

1.2 Waveform Symmetry Simplifications

Recognizing waveform symmetry drastically simplifies coefficient computation:

Symmetry Type Mathematical Condition Harmonic Impact Non-Zero Coefficients

Even Symmetry f ( − t ) = f ( t ) f(-t) = f(t) f ( − t ) = f ( t ) Symmetric about y-axis a 0 ≠ 0 , a n ≠ 0 , b n = 0 a_0 \ne 0, a_n \ne 0, b_n = 0 a 0 ​  = 0 , a n ​  = 0 , b n ​ = 0

Odd Symmetry f ( − t ) = − f ( t ) f(-t) = -f(t) f ( − t ) = − f ( t ) Anti-symmetric about origin a 0 = 0 , a n = 0 , b n ≠ 0 a_0 = 0, a_n = 0, b_n \ne 0 a 0 ​ = 0 , a n ​ = 0 , b n ​  = 0

Half-Wave Symmetry f ( t − T 2 ) = − f ( t ) f\left(t - \frac{T}{2}\right) = -f(t) f ( t − 2 T ​ ) = − f ( t ) Shifted half-cycle inversion Only odd harmonics ( n = 1 , 3 , 5 … n=1,3,5\dots n = 1 , 3 , 5 … ) exist

Quarter-Wave Even Even + Half-Wave Combination b n = 0 b_n = 0 b n ​ = 0 , a n ≠ 0 a_n \ne 0 a n ​  = 0 for odd n n n only

Quarter-Wave Odd Odd + Half-Wave Combination a 0 = a n = 0 a_0=a_n=0 a 0 ​ = a n ​ = 0 , b n ≠ 0 b_n \ne 0 b n ​  = 0 for odd n n n only

1.3 Exponential Fourier Series Form

Using Euler's identity, the Exponential Fourier Series offers a compact representation: f ( t ) = ∑ n = − ∞ ∞ c n e j n ω 0 t f(t) = \sum_{n=-\infty}^{\infty} c_n e^{j n \omega_0 t} f ( t ) = n = − ∞ ∑ ∞ ​ c n ​ e jn ω 0 ​ t

Where complex coefficients c n c_n c n ​ are given by: c n = 1 T ∫ 0 T f ( t ) e − j n ω 0 t d t c_n = \frac{1}{T} \int_{0}^{T} f(t) e^{-j n \omega_0 t} dt c n ​ = T 1 ​ ∫ 0 T ​ f ( t ) e − jn ω 0 ​ t d t

Relationship to trigonometric coefficients: c 0 = a 0 c_0 = a_0 c 0 ​ = a 0 ​ , and for n > 0 n > 0 n > 0 , c n = a n − j b n 2 c_n = \frac{a_n - j b_n}{2} c n ​ = 2 a n ​ − j b n ​ ​ .

2. Continuous-Time Fourier Transform (CTFT)

For non-periodic continuous signals ( T → ∞ T \to \infty T → ∞ ), the discrete line spectrum becomes a continuous spectrum governed by the Continuous-Time Fourier Transform (CTFT) pair: Forward CTFT: F ( ω ) = F { f ( t ) } = ∫ − ∞ ∞ f ( t ) e − j ω t d t \text{Forward CTFT: } F(\omega) = \mathcal{F}\{f(t)\} = \int_{-\infty}^{\infty} f(t) e^{-j\omega t} dt Forward CTFT: F ( ω ) = F { f ( t )} = ∫ − ∞ ∞ ​ f ( t ) e − jω t d t Inverse CTFT: f ( t ) = F − 1 { F ( ω ) } = 1 2 π ∫ − ∞ ∞ F ( ω ) e j ω t d ω \text{Inverse CTFT: } f(t) = \mathcal{F}^{-1}\{F(\omega)\} = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{j\omega t} d\omega Inverse CTFT: f ( t ) = F − 1 { F ( ω )} = 2 π 1 ​ ∫ − ∞ ∞ ​ F ( ω ) e jω t d ω

2.1 Essential Properties of the Fourier Transform

Linearity : F { a f 1 ( t ) + b f 2 ( t ) } = a F 1 ( ω ) + b F 2 ( ω ) \mathcal{F}\{a f_1(t) + b f_2(t)\} = a F_1(\omega) + b F_2(\omega) F { a f 1 ​ ( t ) + b f 2 ​ ( t )} = a F 1 ​ ( ω ) + b F 2 ​ ( ω )

Time Scaling : F { f ( a t ) } = 1 ∣ a ∣ F ( ω a ) \mathcal{F}\{f(at)\} = \frac{1}{|a|} F\left(\frac{\omega}{a}\right) F { f ( a t )} = ∣ a ∣ 1 ​ F ( a ω ​ )

Frequency Shifting : F { f ( t ) e j ω 0 t } = F ( ω − ω 0 ) \mathcal{F}\{f(t) e^{j\omega_0 t}\} = F(\omega - \omega_0) F { f ( t ) e j ω 0 ​ t } = F ( ω − ω 0 ​ )

Duality : If F { f ( t ) } = F ( ω ) \mathcal{F}\{f(t)\} = F(\omega) F { f ( t )} = F ( ω ) , then F { F ( t ) } = 2 π f ( − ω ) \mathcal{F}\{F(t)\} = 2\pi f(-\omega) F { F ( t )} = 2 π f ( − ω )

Time Convolution Theorem : F { f ( t ) ∗ g ( t ) } = F ( ω ) ⋅ G ( ω ) \mathcal{F}\{f(t) * g(t)\} = F(\omega) \cdot G(\omega) F { f ( t ) ∗ g ( t )} = F ( ω ) ⋅ G ( ω )

Frequency Convolution Theorem : F { f ( t ) ⋅ g ( t ) } = 1 2 π [ F ( ω ) ∗ G ( ω ) ] \mathcal{F}\{f(t) \cdot g(t)\} = \frac{1}{2\pi} [F(\omega) * G(\omega)] F { f ( t ) ⋅ g ( t )} = 2 π 1 ​ [ F ( ω ) ∗ G ( ω )]

Parseval's Theorem : Energy equivalence between domains: E = ∫ − ∞ ∞ ∣ f ( t ) ∣ 2 d t = 1 2 π ∫ − ∞ ∞ ∣ F ( ω ) ∣ 2 d ω E = \int_{-\infty}^{\infty} |f(t)|^2 dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |F(\omega)|^2 d\omega E = ∫ − ∞ ∞ ​ ∣ f ( t ) ∣ 2 d t = 2 π 1 ​ ∫ − ∞ ∞ ​ ∣ F ( ω ) ∣ 2 d ω

3. Z-Transform & Discrete-Time Analysis

The Z-Transform is the discrete-time counterpart of the Laplace transform, fundamental to Digital Signal Processing (DSP) and sampled-data control systems.

3.1 Unilateral and Bilateral Z-Transforms

For a discrete sequence x [ n ] x[n] x [ n ] , the bilateral Z-transform is defined as: X ( z ) = Z { x [ n ] } = ∑ n = − ∞ ∞ x [ n ] z − n X(z) = \mathcal{Z}\{x[n]\} = \sum_{n=-\infty}^{\infty} x[n] z^{-n} X ( z ) = Z { x [ n ]} = n = − ∞ ∑ ∞ ​ x [ n ] z − n

Where z = r e j Ω z = r e^{j\Omega} z = r e j Ω is a complex variable in the z z z -plane.

3.2 Region of Convergence (ROC)

The Region of Convergence (ROC) specifies the set of z z z values for which the summation converges absolutely ( ∑ ∣ x [ n ] z − n ∣ < ∞ \sum |x[n] z^{-n}| < \infty ∑ ∣ x [ n ] z − n ∣ < ∞ ). ROC Characterization in the Z-Plane

• Causal Sequence Anti-Causal Sequence Stable LTI System
• +---------------+ +---------------+ +---------------+
• | |z| > |r_max| | | |z| < |r_min| | | Includes Unit |
• | Exterior of | | Interior of | | Circle (|z|=1)|
• | Outer Pole | | Inner Pole | | in the ROC |
• +---------------+ +---------------+ +---------------+
• 3.3 Essential Z-Transform Pairs

Sequence Description Discrete-Time x [ n ] x[n] x [ n ] Z-Domain X ( z ) X(z) X ( z ) Region of Convergence (ROC)

Discrete Impulse δ [ n ] \delta[n] δ [ n ] 1 1 1 Entire z z z -plane

Causal Step u [ n ] u[n] u [ n ] 1 1 − z − 1 = z z − 1 \frac{1}{1 - z^{-1}} = \frac{z}{z - 1} 1 − z − 1 1 ​ = z − 1 z ​ ∣ z ∣ > 1 \vert z\vert> 1 ∣ z ∣ > 1

Causal Exponential a n u [ n ] a^n u[n] a n u [ n ] 1 1 − a z − 1 = z z − a \frac{1}{1 - a z^{-1}} = \frac{z}{z - a} 1 − a z − 1 1 ​ = z − a z ​ ∣ z ∣ > ∣ a ∣ \vert z\vert> \vert a\vert ∣ z ∣ > ∣ a ∣

Anti-Causal Exponential − a n u [ − n − 1 ] -a^n u[-n-1] − a n u [ − n − 1 ] z z − a \frac{z}{z - a} z − a z ​ ∣ z ∣ < ∣ a ∣ \vert z\vert< \vert a\vert ∣ z ∣ < ∣ a ∣

Ramp Weighted n a n u [ n ] n a^n u[n] n a n u [ n ] a z − 1 ( 1 − a z − 1 ) 2 = a z ( z − a ) 2 \frac{a z^{-1}}{(1 - a z^{-1})^2} = \frac{a z}{(z - a)^2} ( 1 − a z − 1 ) 2 a z − 1 ​ = ( z − a ) 2 a z ​ ∣ z ∣ > ∣ a ∣ \vert z\vert> \vert a\vert ∣ z ∣ > ∣ a ∣

4. Difference Equations & Discrete System Stability

Linear constant-coefficient difference equations (LCCDE) model discrete-time systems: ∑ k = 0 N a k y [ n − k ] = ∑ m = 0 M b m x [ n − m ] \sum_{k=0}^{N} a_k y[n-k] = \sum_{m=0}^{M} b_m x[n-m] k = 0 ∑ N ​ a k ​ y [ n − k ] = m = 0 ∑ M ​ b m ​ x [ n − m ]

Taking the Z-transform yields the system transfer function H ( z ) H(z) H ( z ) : H ( z ) = Y ( z ) X ( z ) = ∑ m = 0 M b m z − m ∑ k = 0 N a k z − k H(z) = \frac{Y(z)}{X(z)} = \frac{\sum_{m=0}^{M} b_m z^{-m}}{\sum_{k=0}^{N} a_k z^{-k}} H ( z ) = X ( z ) Y ( z ) ​ = ∑ k = 0 N ​ a k ​ z − k ∑ m = 0 M ​ b m ​ z − m ​

BIBO Stability Criterion : A causal discrete-time LTI system is Bounded-Input Bounded-Output (BIBO) stable if and only if all poles of H ( z ) H(z) H ( z ) lie strictly inside the unit circle in the z z z -plane ( ∣ p i ∣ < 1 |p_i| < 1 ∣ p i ​ ∣ < 1 ).

• 5. PRC ECE Board Exam Worked Examples
• Worked Example 1: Z-Transform & ROC Determination

Problem: Find the Z-transform X ( z ) X(z) X ( z ) and ROC for x [ n ] = 3 ( 0.5 ) n u [ n ] − 2 ( 0.8 ) n u [ n ] x[n] = 3(0.5)^n u[n] - 2(0.8)^n u[n] x [ n ] = 3 ( 0.5 ) n u [ n ] − 2 ( 0.8 ) n u [ n ] .

Solution:

Apply linearity and standard transform pairs: X ( z ) = 3 ⋅ z z − 0.5 − 2 ⋅ z z − 0.8 X(z) = 3 \cdot \frac{z}{z - 0.5} - 2 \cdot \frac{z}{z - 0.8} X ( z ) = 3 ⋅ z − 0.5 z ​ − 2 ⋅ z − 0.8 z ​

Determine individual ROCs:

• For 3 ( 0.5 ) n u [ n ] ⟹ ∣ z ∣ > 0.5 3(0.5)^n u[n] \implies |z| > 0.5 3 ( 0.5 ) n u [ n ] ⟹ ∣ z ∣ > 0.5
• For 2 ( 0.8 ) n u [ n ] ⟹ ∣ z ∣ > 0.8 2(0.8)^n u[n] \implies |z| > 0.8 2 ( 0.8 ) n u [ n ] ⟹ ∣ z ∣ > 0.8

Find the combined ROC (intersection): ROC = { z : ∣ z ∣ > 0.5 } ∩ { z : ∣ z ∣ > 0.8 } ⟹ ∣ z ∣ > 0.8 \text{ROC} = \{z: |z| > 0.5\} \cap \{z: |z| > 0.8\} \implies |z| > 0.8 ROC = { z : ∣ z ∣ > 0.5 } ∩ { z : ∣ z ∣ > 0.8 } ⟹ ∣ z ∣ > 0.8

Simplify X ( z ) X(z) X ( z ) into a single fraction: X ( z ) = 3 z ( z − 0.8 ) − 2 z ( z − 0.5 ) ( z − 0.5 ) ( z − 0.8 ) = 3 z 2 − 2.4 z − 2 z 2 + z ( z − 0.5 ) ( z − 0.8 ) = z 2 − 1.4 z ( z − 0.5 ) ( z − 0.8 ) X(z) = \frac{3z(z - 0.8) - 2z(z - 0.5)}{(z - 0.5)(z - 0.8)} = \frac{3z^2 - 2.4z - 2z^2 + z}{(z - 0.5)(z - 0.8)} = \frac{z^2 - 1.4z}{(z - 0.5)(z - 0.8)} X ( z ) = ( z − 0.5 ) ( z − 0.8 ) 3 z ( z − 0.8 ) − 2 z ( z − 0.5 ) ​ = ( z − 0.5 ) ( z − 0.8 ) 3 z 2 − 2.4 z − 2 z 2 + z ​ = ( z − 0.5 ) ( z − 0.8 ) z 2 − 1.4 z ​

Worked Example 2: Discrete System Stability Evaluation

Problem: A discrete filter has H ( z ) = z z 2 − 0.6 z + 0.25 H(z) = \frac{z}{z^2 - 0.6z + 0.25} H ( z ) = z 2 − 0.6 z + 0.25 z ​ . Determine if the system is stable.

Solution:

Find system poles by setting denominator to zero: z 2 − 0.6 z + 0.25 = 0 z^2 - 0.6z + 0.25 = 0 z 2 − 0.6 z + 0.25 = 0 .

Use quadratic formula: z = 0.6 ± ( − 0.6 ) 2 − 4 ( 1 ) ( 0.25 ) 2 = 0.6 ± 0.36 − 1.0 2 = 0.3 ± j 0.4 z = \frac{0.6 \pm \sqrt{(-0.6)^2 - 4(1)(0.25)}}{2} = \frac{0.6 \pm \sqrt{0.36 - 1.0}}{2} = 0.3 \pm j 0.4 z = 2 0.6 ± ( − 0.6 ) 2 − 4 ( 1 ) ( 0.25 ) ​ ​ = 2 0.6 ± 0.36 − 1.0 ​ ​ = 0.3 ± j 0.4

Calculate the magnitude of the complex poles ∣ z p ∣ |z_p| ∣ z p ​ ∣ : ∣ z p ∣ = ( 0.3 ) 2 + ( 0.4 ) 2 = 0.09 + 0.16 = 0.25 = 0.5 |z_p| = \sqrt{(0.3)^2 + (0.4)^2} = \sqrt{0.09 + 0.16} = \sqrt{0.25} = 0.5 ∣ z p ​ ∣ = ( 0.3 ) 2 + ( 0.4 ) 2 ​ = 0.09 + 0.16 ​ = 0.25 ​ = 0.5

Check stability: Since ∣ z p ∣ = 0.5 < 1.0 |z_p| = 0.5 < 1.0 ∣ z p ​ ∣ = 0.5 < 1.0 , all poles lie strictly inside the unit circle. The filter is BIBO stable .