5.1 Discrete-Time Fourier Transform

离散时间傅里叶变换 (DTFT)

DTFT Explorer 离散时间傅里叶变换探索

Pick a discrete-time signal from the canonical pairs. The right plots show its DTFT magnitude and phase over — note the 2π-periodicity (the shaded band marks one fundamental period).
从典型变换对中选一个离散信号；右侧显示其 DTFT 幅度和相位。注意频谱以 2π 为周期。

DTFT:

X(Ω):

Key fact 关键: DTFT is always 2π-periodic in Ω, so the shaded band fully determines the spectrum. DTFT 在 Ω 上以 2π 为周期，主周期 [−π, π] 已包含全部信息。

Time Domain 时域

Magnitude 幅度谱

Phase 相位谱

Think About It 想一想

Q1: Why is the DTFT 2π-periodic in Ω, but the CTFT is not?为什么 DTFT 以 2π 为周期，而 CTFT 没有周期性？

The discrete frequency is normalized by the sampling period. Adding 2π to Ω corresponds to adding to the analog frequency, which is exactly the sampling rate. Sampling makes the spectrum repeat every — so in the normalized Ω axis the period is exactly 2π.

离散频率 Ω = ωT 已用采样周期归一化。Ω 增加 2π 等同于模拟频率增加一个采样率 ωs，而采样使频谱在每个 ωs 上重复，所以在 Ω 轴上周期就是 2π。

Q2: Try δ[n − c] — the magnitude is flat 1 but the phase is a line. Why?δ[n−c] 的幅度始终为 1，相位却是直线，为什么？

Time shift ⇔ phase ramp: . The magnitude is unchanged (energy unchanged), and the phase is the linear ramp whose slope is the shift amount.

时移对应相位斜坡：δ[n−c] ↔ e^−jΩc。幅度恒为 1（能量不变），相位是斜率为 −c 的直线，斜率即是位移量。

Q3: For aⁿu[n] try a = +0.9 vs a = −0.9. Why does the spectrum peak shift from Ω = 0 to Ω = π?aⁿu[n] 取 a=+0.9 与 a=−0.9，谱峰为何从 Ω=0 移到 Ω=π？

. The denominator is smallest when : for a > 0 this happens at Ω = 0 (slow decay → low-pass), for a < 0 this happens at Ω = π (fast oscillating → high-pass). Sign of a flips smooth ↔ alternating, which is exactly low ↔ high frequency in DT.

分母在 a·e^−jΩ ≈ 1 处最小：a>0 时该点为 Ω=0（缓慢衰减 → 低通）， a<0 时为 Ω=π（正负交替 → 高通）。a 的符号决定信号是平滑还是震荡，对应的 DTFT 即为低频或高频强。

Q4: For the rectangular pulse, why does the spectrum become a Dirichlet kernel (sinc-like)?矩形脉冲为什么得到 Dirichlet 核（类 sinc）频谱？

Geometric sum: . The magnitude is the discrete-time analogue of sinc: nulls at , main-lobe width . Increasing L narrows the main lobe (better frequency resolution) just like CTFT.

几何级数求和给出 Dirichlet 核：零点在 Ω = 2πk/L，主瓣宽度 ≈ 4π/L。L 越大主瓣越窄（频率分辨率越好），与 CTFT 中矩形脉冲变 sinc 的规律一致。

Q5: For windowed cos(Ω₀·n), the spectrum is two narrow peaks at ±Ω₀ — what is the corresponding "ideal" non-windowed result?cos(Ω₀n) 加窗后频谱在 ±Ω₀ 处有窄峰；不加窗的理想结果是什么？

The unwindowed cosine has DTFT — two impulse trains, repeating every 2π. Truncating with a finite rectangular window convolves these with a Dirichlet kernel, producing the visible main lobes plus side-lobes (spectral leakage). This is the discrete-time view of the leakage we saw in §3.4.

理想 cos 的 DTFT 是 ±Ω₀ 处的冲激串（每 2π 重复一次）。截断加矩形窗会与 Dirichlet 核卷积，产生看到的主瓣和旁瓣，即频谱泄漏（与 §3.4 中相同现象）。