MIT OpenCourseWare http://ocw.mit.edu 2.161 Signal Processing: Continuous and Discrete Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.161 Signal Processing - Continuous and Discrete Sampling and the Discrete Fourier Transform 1 1 Sampling Consider a continuous function f(t) that is limited in extent, T1 ≤ t < T2. In order to process this function in the computer it must be sampled and represented by a finite set of numbers. The most common sampling scheme is to use a fixed sampling interval ΔT and to form a sequence of length N: {fn} (n = 0 . . . N − 1), where fn = f(T1 + nΔT ). In subsequent processing the function f(t) is represented by the finite sequence {fn} and the sampling interval ΔT . In practice, sampling occurs in the time domain by the use of an analog-digital (A/D) converter. The mathematical operation of sampling (not to be confused with the physics of sampling) is most commonly described as a multiplicative operation, in which f(t) is multiplied by a Dirac comb sampling function s(t; ΔT ), consisting of a set of delayed Dirac delta functions: ∞ s(t; ΔT ) = � δ(t − nΔT ). (1) n=−∞ We denote the sampled waveform f�(t) as ∞f�(t) = s(t; ΔT )f(t) = � f(t)δ(t − nΔT ) (2) n=−∞ as shown in Fig. 1. Note that f�(t) is a set of delayed weighted delta functions, and that the waveform must be interpreted in the distribution sense by the strength (or area) of each component impulse. The implied process to produce the discrete sample sequence {fn} is by integration across each impulse, that is � nΔT + � nΔT + ∞fn = f�(t)dt = � f(t)δ(t − nΔT )dt (3) nΔT − nΔT − n=−∞ or fn = f(nΔT ) (4) by the sifting property of δ(t). 1.0.1 Spectrum of the Sampled Waveform f�(t): Notice that sampling comb function s(t; ΔT ) is periodic and is therefore describ ed by a Fourier series: 1 ∞ jnΩ0t s(t; ΔT ) = � eΔT n=−∞ 1D. Rowell October 2, 2008 1f ( t )ts ( t ;D T)tf ( t ) = f ( x ) s ( t ;D T)*tFigure 1: Sampling by a comb impulse function. The function f(t) is multiplied by the sampling function s(t; ΔT ) to produce the sampled waveform f�(t). where all the Fourier coefficients are equal to (1/ΔT ). Using this form, the spectrum of the sampled waveform f�(t) may be written � ∞F �(jΩ) = f�(t)e−jΩtdt −∞1 ∞ � ∞ jnΩ0t= � f(t)e e−jΩtdtΔT n=−∞ −∞ 1 ∞ = � F (j (Ω − nΩ0))ΔT n=−∞ 1 ∞ � 2πn�� = � F j �Ω − (5)ΔT ΔT n=−∞ where Ω0 = 2π/ΔT . This tells us that the Fourier transform of a sampled function f�(t) is periodic in the frequency domain with period Ω0, and is a superposition of an infinite number of shifted Fourier transforms F (jΩ) of the original function, scaled by a factor of 1/ΔT . 1.1 The Nyquist Sampling Theorem Given a set of samples {fn} and its generating function f(t), an important question to ask is whether the sample set uniquely defines the function that generated it? In other words, given {fn} can we unambiguously reconstruct f (t)? The answer is clearly no, as shown in Fig. 2 where there are obviously many functions that will generate the given set of samples. In fact there are an infinity of candidate functions that will generate the same sample set. 2tf ( t )Figure 2: Demonstration that a set of samples does not uniquely define a continuous function. There are clearly many functions that would generate the same set of samples. The Nyquist sampling theorem places restrictions on the candidate functions and, if satisfied, will uniquely define the function that generated a given set of samples. The theorem may be stated in many equivalent ways, we present three of them here to illustrate different aspects of the theorem: A function f(t), sampled at equal intervals ΔT , can not be unambiguously reconstructed • from its sample set {fn} unless it is known a-priori that f(t) contains no spectral energy at or above a frequency of π/ΔT radians/s. In order to uniquely represent a function f (t) by a set of samples, the sampling interval ΔT• must be sufficiently small to capture more than two samples per cycle of the highest frequency component present in f(t). There is only one function f(t) that is band-limited to below π/ΔT radians/s that is satisfied • by a given set of samples {fn}. Note that the sampling rate, 1/ΔT , must be greater than twice the highest cyclic frequency fmax in f(t). Thus if the frequency content of f(t) is limited to Ωmax radians/s (or fmax cycles/s) the sampling interval ΔT must be chosen so that πΔT < Ωmax or equivalently 1ΔT < 2fmax The minimum sampling rate to satisfy the sampling theorem fmin = Ωmax/π samples/s is known as the Nyquist rate. 1.2 Aliasing Consider a sinusoid f(t) = A sin(at + φ) sampled at intervals ΔT , so that the sample set is {fn} = {A sin(anΔT + φ)} , and noting that sin(t) = sin(t + 2kπ) for any integer k, �� 2πm� �fn = A sin(anΔT + φ) = A sin a + nΔT + φ (6)ΔT where m is an integer, giving the following important result: 3Given a sampling interval of ΔT , sinusoidal components with an angular frequency a and a + 2πm/ΔT , for any integer m, will generate the same sample set. Figure 3 shows a sinusoid sampled at three different rates. In Fig. 3(a) the waveform is sampled at a rate above the Nyquist rate, and the function is uniquely defied by the samples. In (b) the sampling interval ΔT is at the Nyquist rate (two samples/cycle) and the sample set is ambiguous; note that the function f(t) = 0 generates the same samples. In (c) The sinusoid is undersampled and a lower frequency sinusoid, shown as a dashed line, also satisfies the sample set. The phenomenon demonstrated in Fig. 3 is known as aliasing. After sampling any spectral component in F (jΩ) above the Nyquist frequency π/ΔT will “masquerade” as a lower frequency component within the reconstruction bandwidth, thus creating an erroneous reconstructed function. The phenomenon is also known as frequency folding since the high frequency components will be “folded” down into the assumed system bandwidth. One-half of the sampling