Amplitude-shift keying

Amplitude-shift keying (ASK) is a form of modulation that represents digital data as variations in the amplitude of a carrier wave.

The amplitude of an analog carrier signal varies in accordance with the bit stream (modulating signal), keeping frequency and phase constant. The level of amplitude can be used to represent binary logic 0s and 1s. We can think of a carrier signal as an ON or OFF switch. In the modulated signal, logic 0 is represented by the absence of a carrier, thus giving OFF/ON keying operation and hence the name given.



The simplest and most common form of ASK operates as a switch, using the presence of a carrier wave to indicate a binary one and its absence to indicate a binary zero. This type of modulation is called on-off keying, and is used at radio frequencies to transmit Morse code (referred to as continuous wave operation).


Here is a diagram showing the ideal model for a transmission system using an ASK modulation:


  • ht(t) is the carrier signal for the transmission
  • hc(t) is the impulse response of the channel
  • n(t) is the noise introduced by the channel
  • hr(t) is the filter at the receiver
  • L is the number of levels that are used for transmission
  • Ts is the time between the generation of two symbols


v_i = frac{2 A}{L-1} i - A; quad i = 0,1,dots, L-1

the difference between one voltage and the other is:

Delta = frac{2 A}{L - 1}



s (t) = sum_{n = -infty}^{infty} v[n] cdot h_t (t - n T_s)


z(t) = n_r (t) + sum_{n = -infty}^{infty} v[n] cdot g (t - n T_s)

where we use the notation:

n_r (t) = n(t) * h_r (t)

g(t) = h_t (t) * h_c (t) * h_r (t)


z[k] = n_r [k] + v[k] g[0] + sum_{n neq k} v[n] g[k-n]

In this relationship, the second term represents the symbol to be extracted. The others are unwanted: the first one is the effect of noise, the second one is due to the intersymbol interference.



the transmission will be affected only by noise.

Probability of error

The probability density function to make an error after a certain symbol has been sent can be modelled by a Gaussian function; the mean value will be the relative sent value, and its variance will be given by:

sigma_N = int_{-infty}^{+infty} Phi_N (f) cdot |H_r (f)|^2 df

where Phi_N (f) is the spectral density of the noise within the band and Hr (f) is the continuous Fourier transform of the impulse response of the filter hr (f).

The possibility to make an error is given by:

P_e = P_{e/H_0} cdot P_{H_0} + P_{e/H_1} cdot P_{H_1} + dots + P_{e/H_{L-1}} cdot P_{H_{L-1}}

where P_{e/H_0} is the conditional probability of making an error after a symbol vi has been sent and P_{H_0} is the probability of sending a symbol v0.


P_{H_i} = frac{1}{L}

If we represent all the probability density functions on the same plot against the possible value of the voltage to be transmitted, we get a picture like this (the particular case of L=4 is shown):

The possibility of making an error after a single symbol has been sent is the area of the Gaussian function falling under the other ones. It is shown in cyan just for one of them. If we call P+ the area under one side of the Gaussian, the sum of all the areas will be: 2 L P^+ - 2 P^+. The total probability of making an error can be expressed in the form:

P_e = 2 left(1 - frac{1}{L} right) P^+

We have now to calculate the value of P+. In order to do that, we can move the origin of the reference wherever we want: the area below the function will not change. We are in a situation like the one shown in the following picture:


P^+ = int_{frac{A g(0)}{L-1}}^{infty} frac{1}{sqrt{2 pi} sigma_N} e^{-frac{x^2}{2 sigma_N^2}} d x
= frac{1}{2} operatorname{erfc} left(frac{A g(0)}{sqrt{2} (L-1) sigma_N} right)

where erfc() is the complementary error function. Putting all these results together, the probability to make an error is:

P_e = left(1 - frac{1}{L} right) operatorname{erfc} left(frac{A g(0)}{sqrt{2} (L-1) sigma_N} right)


This relationship is valid when there is no intersymbol interference, i.e. g(t) is a Nyquist function.

See also

External links

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