Time Shifting

Introduction

Time shifting is perhaps the most intuitive signal operation — it moves an entire signal forward or backward along the time axis without altering its shape, amplitude, or frequency content. When you hear an echo in a canyon, you are hearing a time-shifted (delayed) copy of the original sound. When a satellite relays a communication signal, the signal arrives after a propagation delay. When digital data flows through a pipeline register, each stage introduces one clock cycle of delay.

Despite its conceptual simplicity, time shifting has remarkable consequences in the frequency domain. A delay does not change what frequencies are present, but it rotates the phase of every frequency component by an amount proportional to that component's frequency. This linear phase characteristic is fundamental to understanding filter design, group delay, and dispersion.

Mathematical Definition

Continuous-Time

$$y(t) = x(t - t_0)$$

• $t_0 > 0$: The signal is delayed — shifted to the right by $t_0$ seconds
• $t_0 < 0$: The signal is advanced — shifted to the left by $|t_0|$ seconds

The interpretation is straightforward: at any time $t$, the output $y(t)$ equals what the input was $t_0$ seconds earlier. The entire waveform slides rigidly along the time axis.

Discrete-Time

$$y[n] = x[n - n_0]$$

The sequence shifts right by $n_0$ samples (for positive $n_0$). Every sample value remains the same; only the time index at which each value appears changes.

Frequency-Domain Effects

Fourier Transform: The Time-Shift Property

$$\mathcal{F}\{x(t - t_0)\} = e^{-j\omega t_0} \cdot X(j\omega)$$

This single equation reveals everything about time shifting in the frequency domain:

Magnitude: $|Y(j\omega)| = |e^{-j\omega t_0}| \cdot |X(j\omega)| = |X(j\omega)|$

The magnitude spectrum is completely unchanged. A delayed signal contains exactly the same frequencies at exactly the same amplitudes as the original.

Phase: $\angle Y(j\omega) = \angle X(j\omega) - \omega t_0$

A linear phase term $-\omega t_0$ is added. The phase decreases linearly with frequency, with slope $-t_0$. This is called linear phase and it corresponds to a pure time delay.

Proof

Starting from the Fourier Transform definition: $$Y(j\omega) = \int_{-\infty}^{\infty} x(t-t_0) e^{-j\omega t} dt$$

Substitute $\tau = t - t_0$ (so $t = \tau + t_0$, $dt = d\tau$): $$Y(j\omega) = \int_{-\infty}^{\infty} x(\tau) e^{-j\omega(\tau + t_0)} d\tau = e^{-j\omega t_0} \int_{-\infty}^{\infty} x(\tau) e^{-j\omega\tau} d\tau = e^{-j\omega t_0} X(j\omega)$$

Laplace Transform

$$\mathcal{L}\{x(t - t_0) \cdot u(t - t_0)\} = e^{-st_0} \cdot X(s)$$

The factor $u(t-t_0)$ ensures we are dealing with the causal delayed signal. Multiplication by $e^{-st_0}$ in the s-domain represents delay. The ROC remains unchanged.

Z-Transform

$$\mathcal{Z}\{x[n - n_0]\} = z^{-n_0} \cdot X(z)$$

Each sample of delay multiplies by $z^{-1}$. A delay of $n_0$ samples corresponds to multiplication by $z^{-n_0}$. This is the most important property of the Z-Transform — it turns delay elements into algebraic multiplication, converting difference equations into polynomials.

Group Delay and Linear Phase

The concept of group delay is defined as: $$\tau_g(\omega) = -\frac{d}{d\omega}\angle H(j\omega)$$

For a pure time delay, $\angle H(j\omega) = -\omega t_0$, giving $\tau_g = t_0$ (constant for all frequencies). This means all frequency components are delayed by the same amount, preserving the signal shape.

When group delay varies with frequency (non-constant), different frequency components arrive at different times, causing dispersion — the signal shape distorts. Linear phase filters (where all frequency components experience the same delay) are thus highly valued in applications where waveform fidelity matters, such as audio processing and data communications.

Graphical Method

To sketch $x(t - t_0)$:

Step 1: Identify all critical features of $x(t)$: peaks, valleys, zero crossings, start/end points, discontinuities.

Step 2: Add $t_0$ to each critical time value. (For a delay, features move right; for an advance, they move left.)

Step 3: Redraw the signal at the new positions. Shape and amplitudes are unchanged.

Example

If $x(t)$ is a rectangular pulse from $t = 1$ to $t = 4$ with amplitude 3:

• $x(t-2)$: pulse from $t = 3$ to $t = 6$, amplitude still 3 (delayed by 2)
• $x(t+1)$: pulse from $t = 0$ to $t = 3$, amplitude still 3 (advanced by 1)

Combined Operations: The Correct Order

For expressions like $x(at - b)$, always factor out the scaling coefficient first: $$x(at - b) = x(a(t - b/a))$$

The signal is scaled by $a$ and shifted by $b/a$ (not $b$!).

Example: $x(2t - 6) = x(2(t-3))$

• First compress by factor 2, then shift right by 3 (not 6!)
• Or: shift by 6, then compress (which maps the shift of 6 to an effective shift of 3)

Physical Realizations of Time Delay

Analog delay lines: Transmission lines, surface acoustic wave (SAW) devices, or bucket-brigade devices provide physical signal delay.

Digital delay: Simply storing samples in memory and reading them out later. A FIFO buffer of length $N$ implements a delay of $N$ samples.

Propagation delay: Electromagnetic waves in cables travel at $v \approx 2/3 \cdot c$, giving delay $\tau = L/v$ where $L$ is cable length.

Processing latency: Any real-time system introduces computational delay between input and output.

Applications

Echo and Reverb: Artificial reverb sums multiple delayed and attenuated copies: $$y(t) = x(t) + \alpha_1 x(t-\tau_1) + \alpha_2 x(t-\tau_2) + \cdots$$

Phased Array Beamforming: Each antenna element receives a delayed version of the same wavefront. By applying compensating delays, the array steers its beam electronically.

Digital Communication: Multipath channels produce multiple delayed copies of the transmitted signal. Equalizers must identify and compensate for these delays.

Correlation and Template Matching: Sliding a template across a signal (correlation) is systematically computing the similarity at every possible time shift.

Key Takeaways

• Time shift by $t_0$: $y(t) = x(t - t_0)$ — positive $t_0$ delays, negative advances
• Shape, amplitude, and frequency content are all preserved — only temporal position changes
• Fourier domain: adds linear phase $e^{-j\omega t_0}$, magnitude unchanged
• Laplace/Z domains: multiply by $e^{-st_0}$ or $z^{-n_0}$ respectively
• $z^{-1}$ is the fundamental unit delay element in digital system design
• Linear phase (constant group delay) means all frequencies are delayed equally
• For combined operations $x(at-b)$, factor to $x(a(t-b/a))$ to find the true shift $b/a$