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Course by: Thad Welch. E-mail the author

# SIGNAL SAMPLING

Module by: Nguyen Huu Phuong. E-mail the author

Analog signals, in general, are continuous in time. In digital signal processing, we do not use the whole analog signal but replace it by its amplitudes taken at regular intervals. This is sampling . The problem is we must sample the signal so that the samples represent correctly the signal, i.e. from the samples we can reconstruct the original analog signal perfectly.

## Sampling of continuous-time signals

Sampling a continuous-time signal turns it into a correspond discrete-time signal so that it can be processed on digital systems. Actually, the sampling is followed by two other operations, quantization and binary encoding . In reality, the analog-to-digital converters (abbreviated ADC or A/D) do all the three steps.

Figure 1 depicts the sampling of a signal at regular interval t=nt=n size 12{t=n} {}TsTs size 12{T rSub { size 8{s} } } {} where n is an integer, positive and negative. This is uniform sampling that we use routinely. Rarely, nonuniform sampling is mentioned. We denote the samples of the signal x(t) as x(t)x(t) size 12{ { {x}} $$t$$ } {} or x(nTs)x(nTs) size 12{x $$ital "nT" rSub { size 8{s} }$$ } {}. Figure 2 shows the sampling process. It turns out that sampling is just a multiplication of the analog signal x(t) with a sampling signal (or function) s(t):

x(t)=x(nTs)=x(t)s(t)x(t)=x(nTs)=x(t)s(t) size 12{ { {x}} $$t$$ =x $$ital "nT" rSub { size 8{s} }$$ =x $$t$$ s $$t$$ } {}
(1)

The sampling signal s(t) is a regular sequence of narrow pulses δt δt MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiabes7aKjaadshaaaa@3884@ of amplitude 1 (Figure 3) when multiplying s(t) with the signal x(t) we obtain the instantaneous values of x(t) which are the samples. An electric switch (Figure 2b) is a way to implement the sampling: When the contact closes in a short time, the signal passes; and when the contact opens, no output signal appears.

The time distance T s T s MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadsfadaWgaaWcbaGaam4Caaqabaaaaa@37E3@ is called sampling interval or sampling period , fs=1/Tsfs=1/Ts size 12{f rSub { size 8{s} } =1/T rSub { size 8{s} } } {} is sampling frequency (Hz or samples/sec), also called sampling rate. The samples were written as x(nTs)x(nTs) size 12{x $$ital "nT" rSub { size 8{s} }$$ } {}but TsTs size 12{T rSub { size 8{s} } } {} is usually taken as 1, hence the samples will be denoted universally, unless otherwise specified, as x(n). The integer n can represent sample, time, space, but we will often call it time index , or just index, or sample.

When looking at Figure 1 and Figure 3 we may ask if the sampling is appropriate, that is the samples are too close or too far away or just right. This is really a big question and will be answered soon. For the time being, let’s examine the sampling of a sinewave (Figure 4) x(t) having period T x T x MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadsfadaWgaaWcbaGaamiEaaqabaaaaa@37E8@ and frequency F x =1/ T x F x =1/ T x MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadAeadaWgaaWcbaGaamiEaaqabaGccqGH9aqpcaaIXaGaai4laiaadsfadaWgaaWcbaGaamiEaaqabaaaaa@3C5A@ at the sampling rate fsfs size 12{f rSub { size 8{s} } } {} . Different authors use different symbols, this cause certain difficulty for readers. The figure shows the same sinewave but with 3 different sampling frequency f s f s MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgadaWgaaWcbaGaam4Caaqabaaaaa@37F5@ . In the first case f s =8 F x f s =8 F x MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgadaWgaaWcbaGaam4CaaqabaGccqGH9aqpcaaI4aGaamOramaaBaaaleaacaWG4baabeaaaaa@3BBB@ , the samples are quite close and represent very well the signal (from the samples we can reconstruct the signal). In the second case f s =4 F x f s =4 F x MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgadaWgaaWcbaGaam4CaaqabaGccqGH9aqpcaaI0aGaamOramaaBaaaleaacaWG4baabeaaaaa@3BB7@ , still the samples can represent the signal (imagine that we connect the successive sample values to get a triangular wave

which is then passed through an analog lowpass filter to smooth out the wave form). In the last case f s =2 F x f s =2 F x MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgadaWgaaWcbaGaam4CaaqabaGccqGH9aqpcaaIYaGaamOramaaBaaaleaacaWG4baabeaaaaa@3BB5@ , the sampling rate is equal twice the signal frequency. This is the critical case: The samples may or may not represent the signal depending on positions of sampling points.

## The sampling theorem

Let’s consider a certain continuous-time signal x(t) rpresenting certain information such as voice. Its magnitude frequency spectrum is assumed to be as in Figure 5a where FM is its maximum frequency.

The signal is sampled by a sequence of narrow pulses δt δt MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiabes7aKjaadshaaaa@3884@ of amplitude 1 as before. The Fourier series expansion (see section ) of this sampling function is

s(t)= δt T +2 δt T m=1 cos2πm f s t s(t)= δt T +2 δt T m=1 cos2πm f s t MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadohacaGGOaGaamiDaiaacMcacqGH9aqpdaWcaaqaaiabes7aKjaadshaaeaacaWGubaaaiabgUcaRiaaikdadaWcaaqaaiabes7aKjaadshaaeaacaWGubaaamaaqahabaGaci4yaiaac+gacaGGZbGaaGOmaiabec8aWjaad2gacaWGMbWaaSbaaSqaaiaadohaaeqaaOGaamiDaaWcbaGaamyBaiabg2da9iaaigdaaeaacqGHEisPa0GaeyyeIuoaaaa@5298@
(2)

Hence the samples are

x ^ (t)=x(t)s(t)= δt T x(t)+2 δt T m=1 x(t)cos2πm f s t x ^ (t)=x(t)s(t)= δt T x(t)+2 δt T m=1 x(t)cos2πm f s t MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaamaaHaaabaGaamiEaaGaayPadaGaaiikaiaadshacaGGPaGaeyypa0JaamiEaiaacIcacaWG0bGaaiykaiaadohacaGGOaGaamiDaiaacMcacqGH9aqpdaWcaaqaaiabes7aKjaadshaaeaacaWGubaaaiaadIhacaGGOaGaamiDaiaacMcacqGHRaWkcaaIYaWaaSaaaeaacqaH0oazcaWG0baabaGaamivaaaadaaeWbqaaiaadIhacaGGOaGaamiDaiaacMcaciGGJbGaai4BaiaacohacaaIYaGaeqiWdaNaamyBaiaadAgadaWgaaWcbaGaam4CaaqabaGccaWG0baaleaacaWGTbGaeyypa0JaaGymaaqaaiabg6HiLcqdcqGHris5aaaa@619C@
(3)

Where x ^ (t) x ^ (t) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaamaaHaaabaGaamiEaaGaayPadaGaaiikaiaadshacaGGPaaaaa@39F7@ denotes the samples, its Fourier frequency spectrum is X ^ (F) X ^ (F) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaamaaHaaabaGaamiwaaGaayPadaGaaiikaiaadAeacaGGPaaaaa@39A9@ . Thus the spectrum of the sampled signal consists of that of the analog signal (with a multiplying factor δt/T δt/T MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiabes7aKjaadshacaGGVaGaamivaaaa@3A10@ ) and its shifted versions to ±2 f s ,±3 f s ... ±2 f s ,±3 f s ... MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiabgglaXkaaikdacaWGMbWaaSbaaSqaaiaadohaaeqaaOGaaiilaiaaywW7cqGHXcqScaaIZaGaamOzamaaBaaaleaacaWGZbaabeaakiaac6cacaGGUaGaaiOlaaaa@43C1@ This spectrum can also obtain using the Fourier transform (see section ) instead of the Fourier series.

In Figure 5b the spectrum bands do not overlap so we can recover the analog signal by lowpass filtering the central band, or bandpass filtering any other bands. All the bands contain the same information but at different frequencies . In Figure 5c we still can recover the signal but with a precise filter. In Figure 5d the bands overlap and we are in no way to recover the analog signal. So the limiting case is Figure 5c. From this observation, the sampling theoren states as follows.

In order the samples represent correctly the analog signal, the sampling frequency must be greater than twice the maximum frequency of the analog signal:

f s >2 F M f s >2 F M MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgadaWgaaWcbaGaam4CaaqabaGccqGH+aGpcaaIYaGaamOramaaBaaaleaacaWGnbaabeaaaaa@3B8C@
(4)

The limiting frequency 2 F M F M MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadAeadaWgaaWcbaGaamytaaqabaaaaa@37AF@ is called Nyquist rate , and the central frequency interval (- f s f s MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgadaWgaaWcbaGaam4Caaqabaaaaa@37F5@ /2, f s f s MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgadaWgaaWcbaGaam4Caaqabaaaaa@37F5@ /2) is called the Nyquist interval .

For example if a waveform contains the fundamental frequency of 1 kHz and a second harmonic 2 kHz, then the sampling rate must be greater than 2 x 2 kHz = 4 kHz, say 5 kHz or more. Another example is for the voice in the telephone system. The voice is limited by a high quality analog filter at F M F M MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadAeadaWgaaWcbaGaamytaaqabaaaaa@37AF@ = 3.4 kHz, then the sampling frequency must be greater than 2 x 3.4 = 6.8 kHz, say 8 kHz or more.

In the case of Figure 5d there is a phenomenon called aliasing that will be discussed next.

## Aliasing

We would like to know what happens when the signal is sampled below the Nyquist rate, i.e. the sampling theorem is not satisfied. Look at Figure 6. The low-frequency signal x 1 (t) x 1 (t) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaaWcbaGaaGymaaqabaGccaGGOaGaamiDaiaacMcaaaa@3A26@ is sampled 4 times at S 1 , S 2 , S 3 S 1 , S 2 , S 3 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadofadaWgaaWcbaGaaGymaaqabaGccaGGSaGaaGjbVlaadofadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaaGjbVlaadofadaWgaaWcbaGaaG4maaqabaaaaa@3FB4@ and S 4 S 4 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadofadaWgaaWcbaGaaGinaaqabaaaaa@37A8@ in a period of the signal, i.e f s =4 F x1 f s =4 F x1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgadaWgaaWcbaGaam4CaaqabaGccqGH9aqpcaaI0aGaamOramaaBaaaleaacaWG4bGaaGymaaqabaaaaa@3C72@ . From these samples we would be able to recover x 1 (t) x 1 (t) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaaWcbaGaaGymaaqabaGccaGGOaGaamiDaiaacMcaaaa@3A26@ . For the high-frequency signal x 2 (t) x 2 (t) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaaWcbaGaaGOmaaqabaGccaGGOaGaamiDaiaacMcaaaa@3A27@ there are the same 4 samples S 1 , S 2 , S 3 S 1 , S 2 , S 3 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadofadaWgaaWcbaGaaGymaaqabaGccaGGSaGaaGjbVlaadofadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaaGjbVlaadofadaWgaaWcbaGaaG4maaqabaaaaa@3FB4@ and S 4 S 4 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadofadaWgaaWcbaGaaGinaaqabaaaaa@37A8@ in its 9 cycles, so the sampling frequency is just (4/9) Fx2Fx2 size 12{F rSub { size 8{x2} } } {} i.e. under the Nyquist rate. From these sample points of x 2 (t) x 2 (t) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaaWcbaGaaGOmaaqabaGccaGGOaGaamiDaiaacMcaaaa@3A27@ we will recover x 1 (t) x 1 (t) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaaWcbaGaaGymaaqabaGccaGGOaGaamiDaiaacMcaaaa@3A26@ and not the correct x 2 (t) x 2 (t) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaaWcbaGaaGOmaaqabaGccaGGOaGaamiDaiaacMcaaaa@3A27@ . Thus the high frequency signal when undersampled will be recovered as a low-frequency signal. This phenomenon is called aliasing , and the recovered low frequency, which is false, is called the alias of the original high-frequency signal.

To avoid the aliasing there are two approaches: One is to raise the sampling frequency to satisfy the sampling theorem, the other is to filter off the unecessary high-frequency component from the continuous-time signal. We limit the signal frequency by an effective lowpass filter, called antialiasing prefilter , so that the remained highest frequency is less than half of the intended sampling rate. If the filter is not perfect we must give some allowance. For example in voice processing, if the lowpass filter still allows frequencies above 3,4kHz go through even at small amplitude, the sampling frequency should be 10 kHz or more instead of 8 kHz.

The aliasing phenomenon can be shown mathematically. Let’s consider a complex exponential signal at frequency F which is sampled at interwal TsTs size 12{T rSub { size 8{s} } } {} to yield the samples x(nT ss size 12{ {} rSub { size 8{s} } } {}):

x ( t ) = e j2π Ft x ( t ) = e j2π Ft size 12{x $$t$$ =e rSup { size 8{j2π ital "Ft"} } } {} size 12{ drarrow } {} x ( nT s ) = e j2π FnT s x ( nT s ) = e j2π FnT s size 12{x $$ital "nT" rSub { size 8{s} }$$ =e rSup { size 8{j2π ital "FnT" rSub { size 6{s} } } } } {}

Now consider other signals at frequency F±m f s F±m f s MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadAeacqGHXcqScaWGTbGaamOzamaaBaaaleaacaWGZbaabeaaaaa@3BA0@ , m = 0, 1, 2 … sampled to give x m (nT) x m (nT) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaaWcbaGaamyBaaqabaGccaGGOaGaamOBaiaadsfacaGGPaaaaa@3B30@ :

x m (t)= e j2π(F±m f s ) x m (nT)= e j2π(F±m f s )nT x m (t)= e j2π(F±m f s ) x m (nT)= e j2π(F±m f s )nT MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaaWcbaGaamyBaaqabaGccaGGOaGaamiDaiaacMcacqGH9aqpcaWGLbWaaWbaaSqabeaacaWGQbGaaGOmaiabec8aWjaacIcacaWGgbGaeyySaeRaamyBaiaadAgadaWgaaadbaGaam4CaaqabaWccaGGPaaaaOGaaGjbVlabgkDiElaaysW7caWG4bWaaSbaaSqaaiaad2gaaeqaaOGaaiikaiaad6gacaWGubGaaiykaiabg2da9iaadwgadaahaaWcbeqaaiaadQgacaaIYaGaeqiWdaNaaiikaiaadAeacqGHXcqScaWGTbGaamOzamaaBaaameaacaWGZbaabeaaliaacMcacaWGUbGaamivaaaaaaa@603C@

Because

f s T s =1and e j2πm f s n T s = e j2πmn =1 f s T s =1and e j2πm f s n T s = e j2πmn =1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgadaWgaaWcbaGaam4CaaqabaGccaWGubWaaSbaaSqaaiaadohaaeqaaOGaeyypa0JaaGymaiaaywW7caaMf8UaaGzbVlaadggacaWGUbGaamizaiaaywW7caaMf8UaaGzbVlaadwgadaahaaWcbeqaaiaadQgacaaIYaGaeqiWdaNaamyBaiaadAgadaWgaaadbaGaam4CaaqabaWccaWGUbGaamivamaaBaaameaacaWGZbaabeaaaaGccqGH9aqpcaWGLbWaaWbaaSqabeaacaWGQbGaaGOmaiabec8aWjaad2gacaWGUbaaaOGaeyypa0JaaGymaaaa@5B99@

then

X m (n T s )= e j2π(F+m f s ) = e j2πfn T s e j2πm F s n T s = e j2πFn T s =X(nT) X m (n T s )= e j2π(F+m f s ) = e j2πfn T s e j2πm F s n T s = e j2πFn T s =X(nT) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@6A05@
(5)

This result means that two signals x m (t) x m (t) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaaWcbaGaamyBaaqabaGccaGGOaGaamiDaiaacMcaaaa@3A5D@ and x(t) at different frequencies have the same samples. When we recover the signals from these samples then those signals lie within the Nyquist interval [-fs/2, fs/2] (Figure 5b) are recovered correctly, whereas the signals having frequencies outside the Nyquist interval may be aliased into this interval. In general, for an analog signal of frequency F sampled at the sampling rate fs , first we add and subltract frequencies as follows:

f 0 =F±m f s m=0,1,2,... f 0 =F±m f s m=0,1,2,... MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgadaWgaaWcbaGaaGimaaqabaGccqGH9aqpcaWGgbGaeyySaeRaamyBaiaadAgadaWgaaWcbaGaam4CaaqabaGccaaMf8UaaGzbVlaad2gacqGH9aqpcaaIWaGaaiilaiaaysW7caaIXaGaaiilaiaaysW7caaIYaGaaiilaiaac6cacaGGUaGaaiOlaaaa@4D10@
(6)

and then look for the frequencies lying within the Nyquist interval, they are the reconstructed frequencies.

### Example 1

A signal at frequency 50Hz is sampled at 80Hz. What frequency will be recovered ? Repeat when it is sampled at 120Hz.

Solution

With F = 50Hz, f s =80 f s =80 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgadaWgaaWcbaGaam4CaaqabaGccqGH9aqpcaaI4aGaaGimaaaa@3A81@ Hz, the signal is undersampled (not satisfied the sampling theorem). The Nyquist interval is [-40Hz, 40Hz]. The samples do not only represent the frequency F = 50Hz but all frequencies F±m f s =100±m80,m=0,1,2..., F±m f s =100±m80,m=0,1,2..., MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadAeacqGHXcqScaWGTbGaamOzamaaBaaaleaacaWGZbaabeaakiabg2da9iaaigdacaaIWaGaaGimaiabgglaXkaad2gacaaI4aGaaGimaiaaysW7caGGSaGaaGjbVlaad2gacqGH9aqpcaaIWaGaaiilaiaaysW7caaIXaGaaiilaiaaysW7caaIYaGaaiOlaiaac6cacaGGUaGaaiilaaaa@526E@ i.e. the frequencies

f 0 =50,50±80,50±160,50±240... =50,130,30,210,110,290,190... f 0 =50,50±80,50±160,50±240... =50,130,30,210,110,290,190... MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@778E@

Only the frequency -30Hz lies within the Nyquist interval, then the recovered signal will be -30Hz (30Hz and phase reversal). This signal is the alias of the original signal at 50Hz. Notice that 30Hz is just the difference 80Hz – 50Hz

Now, the sampling frequency is 120Hz, the sampling theorem is statisfied, then the original frequency of 50Hz will be recovered. None of other frequencies f 0 =50±m120=50,170,70,290,190... f 0 =50±m120=50,170,70,290,190... MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgadaWgaaWcbaGaaGimaaqabaGccqGH9aqpcaaI1aGaaGimaiabgglaXkaad2gacaaIXaGaaGOmaiaaicdacqGH9aqpcaaI1aGaaGimaiaacYcacaaMe8UaaGymaiaaiEdacaaIWaGaaiilaiaaysW7cqGHsislcaaI3aGaaGimaiaacYcacaaMe8UaaGOmaiaaiMdacaaIWaGaaiilaiaaysW7cqGHsislcaaIXaGaaGyoaiaaicdacaGGUaGaaiOlaiaac6caaaa@56D6@ … lie in the Nyquist interval [-60hZ, 60Hz], except the original frequency of 50Hz as already known.

### Example 2

A DSP system uses the sampling frequency f s =20kHz f s =20kHz MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgadaWgaaWcbaGaam4CaaqabaGccqGH9aqpcaaIYaGaaGimaiaadUgacaWGibGaamOEaaaa@3D37@ to process audio signal frequency-limited at 10 kHz, but the lowpass filter still alows frequencies up to 30 khz pass through even at small amplitudes. What signal will we get back from the samples?

Solution

For sampling rate f s =20kHz f s =20kHz MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgadaWgaaWcbaGaam4CaaqabaGccqGH9aqpcaaIYaGaaGimaiaadUgacaWGibGaamOEaaaa@3D37@ , the Nyquist interval is [-10kHz, 10kHz]. Thus the audio frequency 0 – 10kHz will be recovered as is. The audio frequency from 10 – 20kHz will be aliased into the frequency range 10 – 0kHz, and the audio the audio frequency from 20 – 30kHz will be aliased into the frequency range 0 – 10kHz. The resulting audio will be distorted due to the superposition of the 3 frequency bands.

We end up this section with the block diagram of the general complete DSP system (Fig.1.20). The digital signal output y(n) from the DSP unit is converted by the digital-to-analog converter (DAC or D/A) back to a coarse analog signal which is then lowpass filtered in the postfilter . The finally reconstructed analog signal x ^ (t) x ^ (t) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaamaaHaaabaGaamiEaaGaayPadaGaaiikaiaadshacaGGPaaaaa@39F7@ is, ideally, the same as the original input x(t).

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