UNIT 1: DISCRETE-TIME SIGNALS AND SYSTEMS

I. FUNDAMENTALS OF DISCRETE-TIME SIGNALS

Representation and Classification of Sequences

A discrete-time signal is a sequence of values $x[n]$, where $n$ is an integer (time index).

Classification:

[!TIP] Exam Focus: Always check symmetry about $$\displaystyle n=0 $$ for even/odd classification. Energy signals are finite-length or decay to zero; power signals are infinite and periodic/constant.

Basic Sequence Operations

Given sequences $x[n]$ and $y[n]$:

1. Addition: $$\displaystyle z[n] = x[n] + y[n] $$

2. Multiplication: $$\displaystyle z[n] = x[n] \cdot y[n] $$

3. Time Shifting (Delay/Advance):

   • $$\displaystyle x[n-n_0] $$ is delayed by $$\displaystyle n_0 $$ (if $$\displaystyle n_0>0 $$).

   • $$\displaystyle x[n+n_0] $$ is advanced by $$\displaystyle n_0 $$.

4. Folding (Time Reversal): $x[-n]$

Example Problem (From Past Paper):

Given: $$\displaystyle g[n] = [-2, 1, 5, 3] $$ for $0 \le n \le 3$ (Note: Length=4) $$\displaystyle c[n] = [3.2, 41, 36, -9.5, 0] $$ for $0 \le n \le 4$ (Length=5)

• Addition: Align indices. For $0 \le n \le 3$: $$\displaystyle g[n]+c[n] = [1.2, 42, 41, -6.5] $$. For $$\displaystyle n=4 $$: $$\displaystyle c[4]=0 $$ only.

• Multiplication: For $0 \le n \le 3$: $$\displaystyle g[n] \cdot c[n] = [-6.4, 41, 180, -28.5] $$. For $$\displaystyle n=4 $$: $0$.

[!TIP] Sequences must be defined over a common $n$ range for operations. Pad with zeros if necessary.

Length of a Discrete-Time Signal

• Finite-length signal: Non-zero over a finite interval $M \le n \le N-1$. Its length is $$\displaystyle L = N - M $$.

• Infinite-length signal: Non-zero over an infinite set of $n$ (e.g., $$\displaystyle x[n] = a^n u[n] $$).

• For operations, the resulting sequence's length is determined by the overlapping non-zero intervals.

Nyquist Rate

• Definition: The minimum sampling rate ($$\displaystyle f_s $$) required to reconstruct a continuous-time signal $$\displaystyle x_a(t) $$ from its samples $$\displaystyle x[n] = x_a(nT_s) $$ without aliasing. It is twice the highest frequency component $$\displaystyle f_{\max} $$ present in $$\displaystyle x_a(t) $$.

$$f_s \ge 2f_{\max}$$

The Nyquist rate is $$\displaystyle f_N = 2f_{\max} $$.

• Relation to Discrete-Time: Once sampled, the discrete-time signal $x[n]$ has a discrete-time frequency $$\displaystyle \omega = 2\pi f / f_s $$. Aliasing occurs if $$\displaystyle f_s < 2f_{\max} $$, causing frequency components to "fold" into $[-\pi, \pi]$.

II. Z-TRANSFORM ANALYSIS

Definition and Properties

Bilateral (Two-sided) z-Transform:

$$X(z) = \mathcal{Z}\{x[n]\} = \sum_{n=-\infty}^{\infty} x[n] z^{-n}$$

Unilateral (One-sided) z-Transform: (Used for causal signals with $$\displaystyle n<0 $$ assumed zero)

$$X(z) = \sum_{n=0}^{\infty} x[n] z^{-n}$$

Key Properties:

Region of Convergence (ROC)

• Definition: Set of $z$ values in the complex plane for which the z-transform sum converges (finite).

• Properties:

  1. ROC is always a ring (or disc) centered at origin: $$\displaystyle r_1 < |z| < r_2 $$.

  2. ROC cannot contain any poles of $X(z)$.

  3. ROC for finite-length sequences (non-zero for $M \le n \le N-1$) is entire $z$-plane except possibly $$\displaystyle z=0 $$ or $$\displaystyle z=\infty $$.

     • If $M \ge 0$ and $N-1 \ge 0$ (causal part), ROC includes $$\displaystyle |z| > 0 $$.

     • If $$\displaystyle M < 0 $$ and $$\displaystyle N-1 < 0 $$ (anti-causal part), ROC includes $$\displaystyle |z| < \infty $$.

     • If $$\displaystyle M < 0 < N-1 $$ (two-sided), ROC is a ring $$\displaystyle r_1 < |z| < r_2 $$.

ROC Determination Summary:

Causal and Anti-Causal Systems

• Causal System: Output depends only on present and past inputs. Impulse response $$\displaystyle h[n] = 0 $$ for $$\displaystyle n < 0 $$.

  • ROC: Exterior of outermost pole: $$\displaystyle |z| > r_{\max} $$.

• Anti-causal System: Output depends only on future inputs. $$\displaystyle h[n] = 0 $$ for $$\displaystyle n > 0 $$.

  • ROC: Interior of innermost pole: $$\displaystyle |z| < r_{\min} $$.

• Non-causal: ROC is a ring $$\displaystyle r_1 < |z| < r_2 $$.

[!TIP] Critical Rule: For a rational $H(z)$ (ratio of polynomials), causality $\iff$ ROC is exterior of the outermost pole. Stability $\iff$ ROC includes the unit circle ($$\displaystyle |z|=1 $$). For a causal and stable system, all poles must lie inside the unit circle.

Z-Transform of Finite-Length Sequences

Given: $$\displaystyle x[n] = \alpha^n $$ for $M \le n \le N-1$, and $0$ otherwise.

$$X(z) = \sum_{n=M}^{N-1} \alpha^n z^{-n} = \sum_{n=M}^{N-1} (\alpha z^{-1})^n$$

This is a finite geometric series.

Let $$\displaystyle a = \alpha z^{-1} $$. Then:

$$X(z) = a^M \frac{1 - a^{N-M}}{1 - a} = \frac{\alpha^M z^{-M} - \alpha^N z^{-N}}{1 - \alpha z^{-1}}$$

ROC: Entire $z$-plane except $$\displaystyle z=0 $$ and/or $$\displaystyle z=\infty $$, depending on $M$ and $N$:

• If $M \ge 0$: ROC includes $$\displaystyle z=\infty $$? No, because of $$\displaystyle z^{-M} $$ term. Actually, for $M \ge 0$, the smallest power of $z$ is $-M \le 0$, so $$\displaystyle z=0 $$ may be a singularity. The ROC is $$\displaystyle 0 < |z| < \infty $$ unless $$\displaystyle \alpha=0 $$.

• More precisely: For finite-length $x[n]$, ROC is all $z$ except $$\displaystyle z=0 $$ and $$\displaystyle z=\infty $$ if the sequence has both past ($$\displaystyle n<0 $$) and future ($$\displaystyle n>0 $$) samples. If it is purely causal ($M \ge 0$), ROC includes $$\displaystyle |z| > 0 $$ (exterior of $$\displaystyle z=0 $$). If purely anti-causal ($$\displaystyle N-1 < 0 $$), ROC includes $$\displaystyle |z| < \infty $$ (interior of $$\displaystyle z=\infty $$).

Properties for Causal Signals

For a causal signal $x[n]$ (i.e., $$\displaystyle x[n]=0 $$ for $$\displaystyle n<0 $$):

1. ROC is exterior of the outermost pole: $$\displaystyle |z| > r_{\max} $$.

2. $X(z)$ is analytic (holomorphic) in this ROC.

3. Initial Value Theorem: If $x[n]$ is causal and $X(z)$ has a ROC $$\displaystyle |z|>r_1 $$, then:

$$x[0] = \lim_{z \to \infty} X(z)$$

4. Final Value Theorem: If all poles of $$\displaystyle (1-z^{-1})X(z) $$ are inside the unit circle (i.e., system stable), then:

$$\lim_{n\to\infty} x[n] = \lim_{z \to 1} (1 - z^{-1}) X(z)$$

III. SYSTEM ANALYSIS USING Z-TRANSFORM

Transfer Function and System Function

For a Linear Time-Invariant (LTI) system with impulse response $h[n]$, the transfer function (system function) is:

$$H(z) = \mathcal{Z}\{h[n]\} = \sum_{n=-\infty}^{\infty} h[n] z^{-n}$$

For a rational system described by a linear constant-coefficient difference equation:

$$\sum_{k=0}^{N} a_k y[n-k] = \sum_{k=0}^{M} b_k x[n-k]$$

Taking z-transform (assuming zero initial conditions):

$$H(z) = \frac{Y(z)}{X(z)} = \frac{\sum_{k=0}^{M} b_k z^{-k}}{\sum_{k=0}^{N} a_k z^{-k}} = \frac{b_0 + b_1 z^{-1} + \dots + b_M z^{-M}}{1 + a_1 z^{-1} + \dots + a_N z^{-N}}$$

$H(z)$ is a rational function of $$\displaystyle z^{-1} $$.

Causality and Stability Conditions

For a rational $H(z)$ with ROC $R$:

• Causality: $\iff$ ROC is the exterior of the outermost pole: $$\displaystyle R = \{z : |z| > r_{\max}\} $$.

• Stability (BIBO): $\iff$ ROC includes the unit circle ($$\displaystyle |z|=1 $$). For causal systems, this means all poles must lie inside the unit circle: $$\displaystyle |p_i| < 1 $$.

• Combined (Causal & Stable): ROC is $$\displaystyle |z| > r_{\max} $$ and $$\displaystyle r_{\max} < 1 $$. Hence, all poles inside unit circle.

[!TIP] Exam Trick: Given $H(z)$ in polynomial form, first find poles. If system is causal, ROC is $$\displaystyle |z| > \max|p_i| $$. Check if $$\displaystyle 1 > \max|p_i| $$ for stability.

Inverse Z-Transform Techniques

1. Power Series Expansion (Long Division): Expand $X(z)$ as $$\displaystyle \sum_{n=-\infty}^{\infty} x[n] z^{-n} $$. Coefficients give $x[n]$. Direct but tedious for closed-form.

2. Partial Fraction Expansion (PFE): For rational $X(z)$.

   • For causal $x[n]$ (ROC $$\displaystyle |z|>r_{\max} $$), expand in terms of $$\displaystyle z^{-1} $$ and use $$\displaystyle \mathcal{Z}^{-1}\{ \frac{A}{1 - a z^{-1}} \} = A a^n u[n] $$.

   • For anti-causal $x[n]$ (ROC $$\displaystyle |z|<r_{\min} $$), use $$\displaystyle \mathcal{Z}^{-1}\{ \frac{A}{1 - a z^{-1}} \} = -A a^n u[-n-1] $$.

3. Contour Integration (Residue Method):

$$x[n] = \frac{1}{2\pi j} \oint_C X(z) z^{n-1} dz$$

where $C$ is a counterclockwise contour in the ROC encircling the origin. $$\displaystyle x[n] = \sum \text{Residues of } X(z)z^{n-1} $$ at poles inside $C$.

IV. DISCRETE FOURIER TRANSFORM (DFT)

Definition and N-Point DFT

For a finite-length sequence $x[n]$, $0 \le n \le N-1$:

• Forward DFT:

$$X[k] = \sum_{n=0}^{N-1} x[n] e^{-j \frac{2\pi}{N} kn}, \quad k = 0, 1, \dots, N-1$$

Often written as $$\displaystyle X[k] = \sum_{n=0}^{N-1} x[n] W_N^{kn} $$ where $$\displaystyle W_N = e^{-j2\pi/N} $$ (twiddle factor).

• Inverse DFT (IDFT):

$$x[n] = \frac{1}{N} \sum_{k=0}^{N-1} X[k] e^{j \frac{2\pi}{N} kn}, \quad n = 0, 1, \dots, N-1$$

Frequency Resolution: $$\displaystyle \Delta f = \frac{f_s}{N} $$ (Hz), where $$\displaystyle f_s $$ is the sampling rate. DFT bins are at $$\displaystyle \omega_k = \frac{2\pi k}{N} $$.

Computation of DFT for Standard Sequences

1. Unit Sample Sequence: $$\displaystyle x[n] = \delta[n] $$, $0 \le n \le N-1$.

   $$\displaystyle X[k] = 1 $$ for all $$\displaystyle k=0,1,...,N-1 $$.

2. Constant Sequence: $$\displaystyle x[n] = 1 $$, $0 \le n \le N-1$.

   $$\displaystyle X[k] = N \delta[k] $$ (i.e., $$\displaystyle X[0]=N $$, $$\displaystyle X[k]=0 $$ for $k \ne 0$).

3. Unit Step: $$\displaystyle x[n] = u[n] $$, $0 \le n \le N-1$.

   $$\displaystyle X[k] = \frac{1}{1 - W_N^k} + \frac{N}{2} \delta[k] $$ (for $k \ne 0$). Special case at $$\displaystyle k=0 $$: $$\displaystyle X[0]=N $$.

Orthogonal Transform Pair

A transform $$\displaystyle \mathbf{X} = \mathbf{T} \mathbf{x} $$ is orthogonal if the transformation matrix $\mathbf{T}$ is orthogonal: $$\displaystyle \mathbf{T}^{-1} = \mathbf{T}^T $$ (or $$\displaystyle \mathbf{T}^H $$ for complex).

• DFT Matrix: $\mathbf{W}$ with elements $$\displaystyle W_{kn} = W_N^{kn} = e^{-j2\pi kn/N} $$.

  $\mathbf{W}$ is unitary up to a scale factor $1/\sqrt{N}$: $$\displaystyle \mathbf{W}^H \mathbf{W} = N \mathbf{I} $$.

  Hence, $$\displaystyle \mathbf{W}^{-1} = \frac{1}{N} \mathbf{W}^H $$, which matches IDFT.

• Energy Preservation (Parseval's Theorem):

$$\sum_{n=0}^{N-1} |x[n]|^2 = \frac{1}{N} \sum_{k=0}^{N-1} |X[k]|^2$$

The total energy in time domain equals total energy in frequency domain (scaled by $1/N$).

Properties of DFT

Assume $$\displaystyle x[n] \leftrightarrow X[k] $$, both of length $N$.

1. Linearity: $$\displaystyle ax[n] + by[n] \leftrightarrow aX[k] + bY[k] $$.

2. Circular Shift: $$\displaystyle x[(n-l)]_N \leftrightarrow W_N^{lk} X[k] $$, where $$\displaystyle [(n)]_N $$ denotes modulo-$N$.

3. Circular Convolution: $$\displaystyle x[n] \circledast y[n] \leftrightarrow X[k] \cdot Y[k] $$.

   • Linear convolution of two length-$N$ sequences requires zero-padding to length $2N-1$ to avoid circular aliasing.

4. Symmetry:

   • If $x[n]$ is real: $$\displaystyle X[k] = X^*[N-k] $$ (conjugate symmetric).

   • If $x[n]$ is real and even: $X[k]$ is real and even.

   • If $x[n]$ is real and odd: $X[k]$ is imaginary and odd.

V. DIGITAL FILTER STRUCTURES

FIR Filter Implementations

FIR (Finite Impulse Response): $$\displaystyle H(z) = \sum_{k=0}^{M} b_k z^{-k} $$ (no denominator).

1. Direct Form (Transversal Form):

   • Structure: $$\displaystyle y[n] = \sum_{k=0}^{M} b_k x[n-k] $$.

   • Requires $M$ multipliers and $M$ delay elements.

   • Most straightforward realization.

2. Cascade Form: Connect multiple FIR sections (often second-order) in series: $$\displaystyle H(z) = H_1(z) H_2(z) \dots $$. Advantages: modular, better numerical properties, easy to adjust individual sections.

IIR Filter Implementations

IIR (Infinite Impulse Response): $$\displaystyle H(z) = \frac{\sum_{k=0}^{M} b_k z^{-k}}{1 + \sum_{k=1}^{N} a_k z^{-k}} $$.

1. Direct Form II (Canonical Form):

   • Minimizes number of delay elements to $\max(M,N)$.

   • Uses a common delay line for both feedforward and feedback parts.

   • Realization Example (from Past Paper):

     Given: $$\displaystyle H(z) = \frac{0.44z^{2} + 0.362z + 0.02}{z^{2} + 0.4z^{2} + 0.18z - 0.2} $$.

     Correction: The denominator likely has a typo (two $$\displaystyle z^2 $$ terms). Assume standard form:

     $$\displaystyle H(z) = \frac{0.44z^{2} + 0.362z + 0.02}{z^{2} + 0.4z + 0.18z - 0.2} $$ is inconsistent.

     Let's assume a typical form: $$\displaystyle H(z) = \frac{b_0 + b_1 z^{-1} + b_2 z^{-2}}{1 + a_1 z^{-1} + a_2 z^{-2}} $$.

     To match the given numerator in powers of $z$, divide numerator and denominator by $$\displaystyle z^2 $$:

     $$\displaystyle H(z) = \frac{0.44 + 0.362 z^{-1} + 0.02 z^{-2}}{1 + 0.4 z^{-1} + 0.18 z^{-1} - 0.2 z^{-2}} $$? Still inconsistent.

     Interpretation for exam: The given expression is in positive powers of z. Convert to negative powers:

     $$\displaystyle H(z) = \frac{0.44 + 0.362 z^{-1} + 0.02 z^{-2}}{1 + 0.4 z^{-1} + 0.18 z^{-1} - 0.2 z^{-2}} $$ is wrong.

     Let's assume denominator is $$\displaystyle z^2 + 0.4z + 0.18z - 0.2 $$? That sums to $$\displaystyle z^2 + 0.58z - 0.2 $$.

     Best approach for answer: State that the denominator polynomial must be corrected. Assume a standard second-order IIR:

     $$\displaystyle H(z) = \frac{b_0 + b_1 z^{-1} + b_2 z^{-2}}{1 + a_1 z^{-1} + a_2 z^{-2}} $$.

     For Direct Form II, draw a diagram with two delay elements in a chain. The feedback path taps from the delays are multiplied by $$\displaystyle a_1, a_2 $$ and summed. The feedforward path taps from the input and delays are multiplied by $$\displaystyle b_0, b_1, b_2 $$ and summed to produce output. The intermediate signal after the first adder is the state.

Computational Complexity Reduction

• Exploiting Symmetry in FIR Coefficients:

  • Linear-phase FIR: $$\displaystyle h[n] = h[M-n] $$ (even symmetry) or $$\displaystyle h[n] = -h[M-n] $$ (odd symmetry).

  • For even symmetry ($M$ even or odd), the number of multiplications can be reduced by about half by combining terms.

  • Example: $$\displaystyle y[n] = \sum_{k=0}^{M} h[k] x[n-k] $$. If $$\displaystyle h[k]=h[M-k] $$, then:

    $$\displaystyle y[n] = h[M/2] x[n-M/2] + 2 \sum_{k=0}^{(M/2)-1} h[k] (x[n-k] + x[n-(M-k)]) $$ for even $M$.

• Lattice Structures: For both FIR and IIR. Advantages: stable by construction (for IIR), good for quantization analysis, modular. More multipliers than direct form but better numerical properties.

VI. DIGITAL FILTER DESIGN METHODS

Window Method (FIR Design)

Procedure:

1. Start with an ideal frequency response $$\displaystyle H_d(e^{j\omega}) $$ (e.g., ideal low-pass with cutoff $$\displaystyle \omega_c $$).

2. Find its infinite-duration impulse response $$\displaystyle h_d[n] $$ via inverse DTFT:

$$h_d[n] = \frac{1}{2\pi} \int_{-\pi}^{\pi} H_d(e^{j\omega}) e^{j\omega n} d\omega$$

(e.g., for ideal LPF: $$\displaystyle h_d[n] = \frac{\omega_c}{\pi} \text{sinc}(\frac{\omega_c n}{\pi}) $$).

3. Window $$\displaystyle h_d[n] $$ to make it finite: $$\displaystyle h[n] = h_d[n] \cdot w[n] $$, where $w[n]$ is a window of length $M+1$.

4. The actual frequency response is the convolution of $$\displaystyle H_d(e^{j\omega}) $$ with the window's DTFT $$\displaystyle W(e^{j\omega}) $$.

Common Windows & Effects:

[!TIP] Trade-off: Narrower main lobe (better transition width) $\implies$ Higher sidelobes (worse stopband attenuation). Choose window based on required stopband attenuation.

Park-McClellan (Equiripple) Method

• Goal: Design optimal linear-phase FIR filters that minimize the maximum error (ripple) in passband and stopband, subject to specified ripple magnitudes $$\displaystyle \delta_p, \delta_s $$.

• Alternation Theorem: The optimal filter (minimax) has an equiripple error characteristic. There exist at least $L+2$ extremal frequencies where the error alternates in sign and reaches $\pm \delta$, where $L$ is the number of filter coefficients (or degrees of freedom).

• Procedure (Remez Exchange Algorithm):

  1. Specify passband/stopband edges, $$\displaystyle \delta_p $$, $$\displaystyle \delta_s $$.

  2. Initialize a set of $L+2$ extremal frequencies.

  3. Solve a linear system to find filter coefficients that satisfy equiripple condition at these frequencies.

  4. Check if the error at all frequencies is within $\delta$. If not, update extremal set and repeat.

• Result: Filter with equal ripples in passband and stopband, providing the minimum possible order $M$ for given specifications.

VII. ADVANCED CONCEPTS IN DSP

Stationarity in Random Processes

A random process $x[n]$ is an infinite collection of random variables indexed by $n$.

• Strict-Sense Stationary (SSS): All statistical properties (e.g., PDF, moments) are invariant to a time shift $$\displaystyle n_0 $$.

  For any $$\displaystyle n_0 $$ and any set $$\displaystyle n_1, n_2, \dots, n_k $$:

$$f_{x[n_1], x[n_2], \dots}(x_1, x_2, \dots) = f_{x[n_1+n_0], x[n_2+n_0], \dots}(x_1, x_2, \dots)$$

• Wide-Sense Stationary (WSS): Weaker condition:

  1. Constant Mean: $$\displaystyle E[x[n]] = \mu_x $$ (independent of $n$).

  2. Autocorrelation depends only on lag: $$\displaystyle R_{xx}[m] = E[x[n] x^*[n-m]] $$ is a function of $m$ only, not $n$.

  • Implication: For WSS, $$\displaystyle R_{xx}[m] $$ is even: $$\displaystyle R_{xx}[m] = R_{xx}^*[-m] $$.

  • Note: WSS does not imply SSS, but SSS implies WSS.

[!TIP] In DSP, we almost always assume WSS for random processes. This allows us to use the power spectral density (PSD) $$\displaystyle S_{xx}(e^{j\omega}) $$, which is the DTFT of $$\displaystyle R_{xx}[m] $$.

Optimal Filtering

• Concept: Design a filter (estimator/predictor) that minimizes an error criterion (usually mean-square error, MSE) based on observed data.

• Wiener Filter: The optimal (MSE) linear filter for estimating a desired signal $d[n]$ from an observed signal $x[n]$, when both are jointly WSS.

  • Filter Types:

    • Filtering (Smoothing): Estimate $d[n]$ from $x[n], x[n-1], \dots$ (causal or non-causal).

    • Prediction: Estimate $d[n+p]$ from $x[n], x[n-1], \dots$ (future prediction).

    • Estimation: Estimate $d[n]$ from $x[n]$ only.

  • Wiener-Hopf Equations: Determine optimal filter coefficients $$\displaystyle h_{opt}[k] $$ by solving:

$$\sum_{k=0}^{N-1} h_{opt}[k] R_{xx}[m-k] = R_{dx}[m], \quad m=0,1,\dots,N-1$$

    In matrix form: $$\displaystyle \mathbf{R} \mathbf{h} = \mathbf{p} $$, where $\mathbf{R}$ is Toeplitz autocorrelation matrix of $x[n]$, $\mathbf{p}$ is cross-correlation vector between $d[n]$ and $x[n]$.

*   **Minimum MSE:** $$\displaystyle MSE_{min} = R_{dd}[0] - \mathbf{p}^H \mathbf{R}^{-1} \mathbf{p} $$.

Symmetry Properties of Discrete-Time Signals

Classification with respect to $$\displaystyle n=0 $$:

1. Even Symmetry: $$\displaystyle x[n] = x[-n] $$.

   • Example: $$\displaystyle x[n] = \cos(\omega_0 n) $$.

   • DTFT is real and even: $$\displaystyle X(e^{j\omega}) = X(e^{-j\omega}) = X^* (e^{j\omega}) $$.

2. Odd Symmetry: $$\displaystyle x[n] = -x[-n] $$.

   • Example: $$\displaystyle x[n] = \sin(\omega_0 n) $$.

   • DTFT is imaginary and odd: $$\displaystyle X(e^{j\omega}) = -X(e^{-j\omega}) = -X^* (e^{j\omega}) $$.

3. Neither: Most signals.

4. Decomposition: Any signal can be written as sum of even and odd parts:

$$x[n] = x_e[n] + x_o[n]$$

where

$$x_e[n] = \frac{1}{2} (x[n] + x[-n]), \quad x_o[n] = \frac{1}{2} (x[n] - x[-n])$$

*   **Even part** is always symmetric.

*   **Odd part** is always antisymmetric.

*   For **real** $x[n]$, $$\displaystyle x_e[n] $$ is real and even, $$\displaystyle x_o[n] $$ is real and odd.

*   For **complex** $x[n]$, no such simplicity.

[!TIP] This decomposition is crucial for understanding linear-phase FIR filters: a linear-phase FIR impulse response has even symmetry (Type I, II) or odd symmetry (Type III, IV).