This work aims to enhance the quality of narrowband (0–4 kHz) voice signal in terms of frequency components, i.e. missing high‐frequency components in a range of 4–8 kHz. The proposed artificial bandwidth extension framework uses the H∞ optimisation. In this context, a signal model is used to get a better representation of wideband (0–8 kHz) information of a signal. The H∞ optimisation is used to obtain the synthesis filter for a given signal model, which is used to synthesise the high‐band (4–8 kHz) signal. The discrete Fourier transform addition is performed to add the narrowband signal and estimated high‐band signal for removing the leaked information from the synthesis filter and non‐ideal low pass filter. Gain adjustment is performed on the estimated high‐band signal to make its energy equal to the true high‐band signal. Non‐stationary characteristics of speech signals generate an assorted variety in synthesis filters and corresponding gain. For this, a deep neural network (DNN) is used to estimate the synthesis filter and gain by using the given narrowband information. The authors analyse the performances of the DNN model on two data sets. Objective and subjective analyses are carried out on these data sets.